Jonathan Tennyson- Resonance Parameters and Quantum Defects for Superexcited H2

8/3/2019 Jonathan Tennyson- Resonance Parameters and Quantum Defects for Superexcited H2

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ATOMIC DATA AND NUCLEAR DATA TABLES 64, 253277 (1996)ARTICLE NO. 0023

RESONANCE PARAMETERS AND QUANTUM DEFECTS

FOR SUPEREXCITED H 2

JONATHAN TENNYSON*

Institute for Theoretical Atomic and Molecular PhysicsHarvardSmithsonian Center for Astrophysics

60 Garden Street, Cambridge, Massachusetts 02138

The results of R-matrix calculations on electron collisions with H /2 are presented. These calculationsinclude the three lowest states of H /2 (2S

/

g , 2S/

u and 2P u ). Positions and widths of the Feshbachresonances converging to both the 2S /u and 2P u states are given, as are autoionization branching ratiosfor resonances lying above the 2S /u state. Complex quantum defects, calculated by performing scatteringcalculations above threshold, are presented for all three states considered. Results are tabulated for 12symmetries ( 1S /g , 1S

/

u , 1P u , 1P g , 1D g , 1D u , 3S/

g , 3S/

u , 3P u , 3P g , 3D g , 3D u ) and 13 internuclear separa-tions from 1 to 4 a 0 . 1996 Academic Press, Inc.

* E-mail: [email protected]. Permanent address: Department of Physics and Astronomy, UniversityCollege London, Gower St., London WC1E6BT, UK

092-640X/96 $18.00

Copyright 1996 by Academic Press, Inc.All rights of reproduction in any form reserved. Atomic Data and Nuclear Data Tables, Vol. 64, No. 2, November 1996253


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CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

EXPLANATION OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

TABLESI. Resonances Converging to the 2S /u State of H

/

2 . . . . . . . . . . . 258II. Resonances Converging to the 2P u State of H

/

2 . . . . . . . . . . . 263III. Quantum Defects for Rydberg States of H 2 . . . . . . . . . . . . . . . 268IV. Complex Quantum Defects at the 2S /u Threshold . . . . . . . . . . 270V. Complex Quantum Defects at the 2P u Threshold . . . . . . . . . . 274

INTRODUCTION

Scope The results are tabulated for 13 internuclear separatiobetween 1 and 4 a 0 .

Processes involving molecular hydrogen are of greatractical and fundamental importance. Superexcited Calculationstates, or resonances, of H 2 play an important role in a

Calculations on electron collisions with H /2 were umber of these processes including dissociative recom-formed using the R-matrix method. The wavefunctionination, 1 photoionization, 2 and dissociative photoioniza-used were developed and tested by Branchett and Tennon. 3 There have been many studies of resonance posi-son 11 for studying transitions to Rydberg states of ons and widths for low-lying members of the resonanceand by Sarpal and Tennyson 12 for calculating vibratioeries which converge to the rst excited, 2S /u state of excitation cross sections.H /2 (see Refs. 47 and references therein).

The present calculations improve on previous R-matrDespite the importance of this system, only Tennysoncalculations of H**2 resonance widths 4,6,7 in that the wnd Noble 4 have systematically studied the resonances of function function expansion includes the lowest three statll low-lying symmetries as a function of H H in-of the target H

/

2 ion with2

S/

g ,2

S/

u , and2

P u symmernuclear separation. These workers used a two-state R- Furthermore, a more exible representation of the targetmatrix model, which has been superseded by more recent orbitals is used to represent not only the 2P u target alculations, and did not consider internuclear separations but also the P polarization effects. Further details of arger than 2.6 a 0 , which is inadequate for modeling dis- calculations can be found in Refs. 11 and 12.ociative processes. A new feature of the present calculations is the pr

A number of recent experiments 1,8,9 have presented ence of resonance series converging to the 2P u stahallenges to the available theoretical data for this system. H /2 . When such resonances lie above the 2S

/

u state, t would therefore appear timely to present a complete can autoionize to either the ground 2S /g or excitednd systematic evaluation of the low-lying resonances of of H /2 . Branching ratios for this were calculated using tuperexcited molecular hydrogen, H**2 . This has been time-delay matrix method of Smith. 13,14one in two ways: by explicitly tting the eigenphase

Resonances were located and tted to a BreitWignums for low-lying resonances in a series and by calculat- form using an automated search procedure. 15 For all rng quantum defects for resonances series by considering nances the time-delay matrix was constructed and diagbove threshold scattering. For resonances converging to nalized at the tted resonance position, E r . The larhe 2P u state of H

/

2 and lying above the 2S/

u state, eigenvalue of this matrix yielded an alternative estimaranching ratios for autoionization have also been calcu- of the resonance width, G, and, where appropriate, ated. These resonances, although observed experimen- corresponding eigenvector gives the branching ratio.ally, 10 have not previously been considered. most cases the two methods of estimating the resonan

The present calculations have been performed using widths gave excellent agreement. Signicant disagrehe R-matrix method for the 1S /g , 1S

/

u , 1P u , 1P g , 1D g , ments resulted only from poor initial ts. These caswere retted. Parameters obtained from these resonancD u , 3S

/

g , 3S/

u , 3P u , 3P g , 3D g , and 3D u total symmetries.

254 Atomic Data and Nuclear Data Tables, Vol. 64, No. 2, November 1996


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ts are tabulated in Tables I and II for resonances which until agreement was obtained between estimates obtainedfrom the tting to the BreitWigner form and the timonverge to the 2S /u and 2P u state of H

/

2 , respectively.delay method. In case of minor disagreements the naComplex quantum defects were obtained from the reso-rower estimate was selected.ances using the formulae

It proved difcult to obtain reliable ts to resonancwhich approached the H /2 ground state, such as the low E r E t 0

1n 2

, G 4 Bn 3

, (1)1S /g resonance at internuclear separation R 2.6 a 0 . problem has been well studied elsewhere, 6 and no spewhere energies are in rydberg (Ry). E t is the energy of effort was made to obtain resonance parameters in the threshold with which the particular (resonance) statecrossing region.

an be associated. The effective quantum number, n , is For the quantum defect parameters, the crucial paramelated to the real part of the quantum defect byter is the energy, e, above threshold at which the S-ma

n n 0 A, (2) is computed. A number of tests were performed beforewas decided that e 0.0008 Ry constituted a reliawhere n is an integer. For bound states, such as the Ryd-value. Again it was found that the real part of the complerg states of H 2 detailed in Table III, B 0. For reso-defect, A, which corresponds to the position was muances, the complex quantum defect, m, is given bymore stable than the imaginary part, B, from which r

m A / 2iB . (3) nance widths can be estimated. A few B values wfound to vary quite strongly as a function of e.Estimates of A and B, which are assumed to vary

The tables of quantum defects label each column wimoothly and usually slowly with n , can be obtained bypartial wave quantum numbers. Of course these quantu

erforming scattering calculations above threshold.16,17

numbers are only approximate and were obtained by ouch calculations were performed at 0.0008 Ry abovedering the quantum defects according to their dominaach threshold. Table III presents estimates of the quan-eigenvectors. In most cases the ( l, m) values presentedum defects of the high n Rydberg states of H 2 . Tablesindeed reect the dominant characteristic of the quantuV and V present complex quantum defects for the highdefects. However in some cases, particularly neresonances converging to the 2S /u and 2P u states of (avoided) crossings, this is not so. These instances aH /2 respectively. It is possible to correlate the threshold characterized by A values which either approach eesonance parameters with those presented for the low nother and move apart again or cross as a function members of the same series presented in Tables I and II.internuclear separation, R.As parameters for only a few resonances are presented

The tabular material presented below may also be on Tables I and II, these generally correlate with the lowesttained in electronic form by anonymous ftp from jonnartial waves, ( l, m) values, given in Tables IV and V.phys.ucl.ac.uk or via the authors world wide web hompage at http://jonny.phys.ucl.ac.uk/home.html. The leAccuracyhave been placed in directory pub/astrodata/H2**, whe

There are two aspects which need to be considered the tables are given in both L ATEX and simple text fowhen attempting to assess the accuracy of the data pre-ented here: the accuracy of the model employed and any Acknowledgmentsrrors in the methods used to extract the parameters.

I thank Annick Suzor-Weiner for encouraging me It is difcult to give a denitive estimate of the accu-re-address this problem. I also thank the staff and visitoacy of the underlying calculations. However there areat the Institute for Theoretical Atomic and Molecumany calculations of H**2 resonance positions and widths,Physics (ITAMP) for many helpful discussions, and nd the present results are in very good agreement withparticular Hossein Sadeghpour for drawing the time-dela

hose obtained in recent calculations.6,7

Indeed, no evi- matrix method to my attention. This work was supportence could be found for errors or omissions in previousby the US National Science Foundation through a graalculations which had been suggested by analysis of for ITAMP at Harvard University and Smithsonian Astroome recent experiments. 8,9physical Observatory and by the UK Engineering aIn estimating resonance parameters, the positions andPhysical Science Research Council.he widths behave somewhat differently. It is easy to

btain reliable resonance energies in a calculation: even Referencesoor ts yield good estimates. Conversely resonance

widths can be very sensitive to the tting procedure used. 1. C. Stromholm et al., Phys. Rev. A 52, R4320 (1Usually poor ts were found to signicantly overestimatehe resonance width. In this work resonances were retted 2. G. Raseev, J. Phys. B: At. Mol. Phys. 18, 423 (1



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3. K. Kirby, T. Uzer, A. C. Allison, and A. Dalgarno, Kouchi, and Y. Hatano, J. Phys. B: At. Mol. OptPhys. 28, L465 (1995)J. Chem. Phys. 75, 2820 (1981); K. P. Kirby, AIP

Conference Proceedings 347, 43 (1995) 10. Z. X. He, J. N. Cutler, S. H. Southworth, L.Hughey, and J. A. R. Samson, J. Chem. Phys.4. J. Tennyson and C. J. Noble, J. Phys. B: At. Mol.

Phys. 18, 155 (1985) 3912 (1995)

11. S. E. Branchett and J. Tennyson, J. Phys. B: At. M5. L. A. Collins, B. I. Schneider, C. J. Noble, C. W.McCurdy, and S. Yabushita, Phys. Rev. Lett. 57, 980 Opt. Phys. 25, 2017 (1992)(1986) 12. B. K. Sarpal and J. Tennyson, Mon. Not. R. A

Soc. 263, 909 (1993)6. I. Shimamura, C. J. Noble, and P. G. Burke, Phys.Rev. A 41, 3545 (1990) 13. F. T. Smith, Phys. Rev. 114, 349 (1960)

7. L. A. Collins, B. I. Schneider, and C. J. Noble, Phys. 14. D. Stibbe and J. Tennyson, J. Phys. B: At. Mol. ORev. A 45, 4610 (1992); L. A. Collins, B. I. Schnei- Phys. 29, 4267 (1996)der, D. L. Lynch, and C. J. Noble, Phys. Rev. A 52,1310 (1995) 15. J. Tennyson and C. J. Noble, Comput. Phys. Comm

33, 421 (1984)8. C. J. Latimer, K. F. Dunn, N. Kouchi, M. A. McDon-ald, V. Srigengan, and J. Geddes, J. Phys. B: At. Mol. 16. M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983)Opt. Phys. 26, L595 (1993) and 27, 2961 (1994) 17. J. Tennyson, J. Phys. B: At. Mol. Opt. Phys. 21,

(1988)9. T. Odagiri, N. Uemura, K. Koyama, M. Ukai, M.



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EXPLANATION OF TABLES

TABLE I. Resonances Converging to the 2S /u State of H /2

TABLE II. Resonances Converging to the 2P u State of H /2

TABLE III. Quantum Defects for Rydberg States of H 2

TABLE IV. Complex Quantum Defects at the 2S /u Threshold

TABLE V. Complex Quantum Defects at the 2P u Threshold

The tables are subdivided according to the total symmetry of the H 2 system.The columns of quantum defect in Tables IIIV are labeled by partial waveparameters ( l, m). As discussed in the introductory text, these designationsare only approximate. Other parameters are dened as:

R Internuclear separation in a 0 , the Bohr radius E r Resonance position in Ry; Eq. (1)G Resonance width in Ry; Eq. (1)n Effective quantum number; Eq. (2) A Real part of the (complex) quantum defect; Eq. (3) B Half the imaginary part of the complex quantum defect; Eq. (3)

b 0 Branching ratio for autoionization to H/

22

S/

gb 1 Branching ratio for autoionization to H

/

22S /u



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TABLE I. Resonances Converging to the 2S /u State of H/

2

See page 257 for Explanation of Tables



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2




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2




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2




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2




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TABLE II. Resonances Converging to the 2P u State of H/

2




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2




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2




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2




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2




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TABLE III. Quantum Defects for Rydberg States of H 2See page 257 for Explanation of Tables



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TABLE III. Quantum Defects for Rydberg States of H 2See page 257 for Explanation of Tables



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TABLE IV. Complex Quantum Defects at the 2S /u ThresholdSee page 257 for Explanation of Tables



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TABLE V. Complex Quantum Defects at the 2P u ThresholdSee page 257 for Explanation of Tables



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Jonathan Tennyson- Resonance Parameters and Quantum Defects for Superexcited H2