Direct Variational Methods for Complex Resonance Energies - ACS

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C. WILLIAM MCCURDY Department of Chemistry, Ohio State University, Columbus, OH 43210 In contrast to e a r l i e r complex coordinate methods which require a s p e c i f i c analytic continuation of the Hamiltonian, complex basis function methods for resonances rely on the existence of a complex v a r i a ­ t i o n a l p r i n c i p l e for complex resonance energies, E (R) - iΓ(R)/2, and are thus considerably more general, f l e x i b l e and successful than their ante­ cedents. A summary i s presented of several methods which have appeared i n the l i t e r a t u r e i n this context with a view to displaying their s i m i l a r i t i e s and common conceptual grounding. Those methods include: (1) complex SCF, (2) complex s t a b i l i z a t i o n (CI), (3) the saddle-point coordinate rotation methods, and (4) analytic continuation of s t a b i l i z a t i o n graphs. The last of these approaches requires only real-valued eigenvalue calculations, but nonetheless yields complex resonance energies d i r e c t l y . r

Recent years have seen the development of a number of methods for direct calculation of complex resonance energies, both i n the context of electron scattering from atoms and molecules and i n the context of heavy-particle scattering. This a r t i c l e i s not intended as a review of that l i t e r a t u r e . Rather, I present here a brief summary of some of our own work together with a description of a few other approaches with the intention of exhibiting the common theme they employ. That theme i s a generalized complex v a r i a t i o n a l p r i n c i p l e for resonances, and the various methods discussed here differ only i n the forms of the trial wavef unctions they prescribe. Before beginning the s p e c i f i c description of these methods I w i l l describe the v a r i a t i o n a l principle and, to some extent, i t s o r i g i n s . 0097-6156/ 84/ 0263-0017S06.00/0 © 1984 American Chemical Society

Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

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Complex Variational Principles for Resonances The method of complex scaling of coordinates, known variously as rotated coordinates, complex coordinates and d i l a t a t i o n a n a l y t i c i t y , i s the predecessor of a l l of the complex v a r i a t i o n a l methods we w i l l consider* Although complex coordinates have long been a textbook device for establishing the analytic structure of the S-matrix i n potential scattering (1), i t was the theorems of Aguilar, Balslev and Combes (_2,_3) on many-particle systems which formed the basis of the f i r s t atomic resonance calculations by this approach. Their now f a m i l i a r results state that i f a l l the coordinates i n an atomic Hamiltonian are scaled according to + 16 (1) the spectrum of the resulting nonhermitian Hamiltonian consists of discrete eigenvalues at the bound-state energies and at the complex energies of resonances, E -ir/2, which have been exposed by the rotation of the continuous spectra associated with each scattering threshold into the complex plane by an angle equal to -26. Moreover the eigenfunctions associated with the complex resonance eigenvalues are square integrable. Figures showing the complex spectrum of the complex scaled Hamiltonian, HQ = H({r^e }), can be found i n several reviews on the subject (4-6). r

The f i r s t computational implementations of these theorems were made by Doolen, Hidalgo, Nuttall and Stagat (7-9) for two-electron atoms. These were configuration interaction (CI) calculations using real valued r a d i a l basis functions and the complex Hamiltonian, HQ• They made use of a complex v a r i a t i o n a l principle of the form 6E m/6V = 0 e

( 2 )

with (

}

JW * H 9

l

J

¥

dx

/fdx

( 3 )

where the notation (Y)^*^ means that only the angular factors (spherical harmonics) i n the wavefunctions are complex conjugated. The variations i n Equation 2 i n these calculations (7-9) were variations i n complex c o e f f i c i e n t s i n an otherwise ordinary CI expansion i n Hylleraas functions. The v a l i d i t y of this approach rests on the fact that the eigenfunctions of HQ i n question are square integrable. Equations 2 and 3 state a complex v a r i a t i o n a l theorem, which has been used and discussed extensively i n the l i t e r a t u r e (4-6,10), but i t i s not the v a r i a t i o n a l theorem on which the most successful methods are based and which concerns us here. The more general complex v a r i a t i o n a l p r i n c i p l e which does concern us makes use of a real-valued Hamiltonian and can be stated simply as 5Em/6Y - 0

Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

( 4 )

2.

MCCURDY

Complex Resonance Energies

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with E [ t ]

.

f W ^ H t d x J

where kp denotes the square-integrable complex "continuum" orbital. Such a c a l c u l a t i o n might suffer even more severe convergence problems than a t y p i c a l multiconfiguration SCF calcu­ l a t i o n on an excited state. On the other hand, i t would be easy to build trial functions i n this approach which should provide excellent approximations to the complex resonance energy. To t h i s author's knowledge, no such c a l c u l a t i o n has yet been attempted. Conclusion In this discussion we have attempted to display the unifying theme of d i r e c t v a r i a t i o n a l calculations on resonances. Since t h i s was not intended to be a review a r t i c l e , we have omitted several important contributions to the subject and apologies are due to those authors whose work i s neglected here. In p a r t i c u l a r work on v a r i a t i o n a l bounds on complex energies, (41,43) i s a very promising development. Our hope has been that by presenting the subject i n this s i m p l i f i e d way we could stimulate further applications of the basic idea, p a r t i c u l a r l y i n heavy-particle resonance problems for which even more successful versions of the analytic continuation of s t a b i l i z a t i o n graphs can surely be developed. Because they have p r a c t i c a l implications we have also emphasized some formal problems which have gone largely neglected i n the l i t e r a t u r e . In p a r t i c u l a r we cannot address the problem of l i n e shapes i n photoionization by adding a background contribution to the resonance contribution to the dipole o s c i l l a t o r strength u n t i l we know for certain how to use our complex v a r i a t i o n a l wavefunctions to do dipole matrix elements. Acknowledgment s Our work i n this area has been supported by the National Science Foundation. I would also l i k e to thank Tom Rescigno, Bobby Junker, B i l l Reinhardt, Barry Simon, and others working i n this area for stimulating conversations which have often set straight my ideas on the subject. Literature Cited 1.

See for example: Taylor, J . R. "Scattering Theory"; John Wiley and Sons: New York, 1972; p. 222.

Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

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McCURDY

2. 3. 4. 5. 6.

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7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

Complex Resonance Energies

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A g u i l a r , J.; Combes, J. Commun, Math. Phys. 1971, 22, 269. B a l s l e v , E.; Combes, J. Commun. Math. Phys. 1971, 22, 280. R e i n h a r d t , W. P. Ann. Rev. Phys. Chem. 1982, 33, 223. McCurdy, C. W. In " A u t o i o n i z a t i o n I I " ; Temkin, A., Ed.; Plenum: New York, 1984 ( t o a p p e a r ) . Simon, B. I n t . J. Quantum Chem. 1978, 14, 529. The i s s u e in which t h i s article appears is d e v o t e d entirely t o complex scalling. Doolen, G. D.; H i d a l g o , M.; N u t t a l l , J . ; S t a g a t , R. W. In "Atomic P h y s i c s " ; Smith, S. J . ; W a l t e r s , G. K., Eds.; Plenum: New York, 1973, p. 257. Doolen, G. D.; Nuttall, J . ; S t a g a t , R. W. Phys. Rev. A 1974, 10, 1612. Doolen, G. D.. J . Phys. B 1975, 8, 525. Moiseyev, N. M o l . Phys. 1982, 47, 585. R e s c i g n o , T. N.; McCurdy, C. W.; O r e l , A. E . . Phys. Rev. A 1978, 17, 1931. J u n k e r , B. R.; Huang, C. L . Phys. Rev. A 1978, 18, 313. Simon, B. Phys. Lett. 1979, 71A, 211. McCurdy, C. W. Phys. Rev. A 1980, 21, 464. H e r r i c k , D. R.; Stillinger, F. H. J . Chem. Phys. 1975, 62, 4360. H e r r i c k , D. R. J. Chem. Phys. 1976, 65, 3529. Sherman, P. R.; H e r r i c k , D. R. Phys. Rev. A 1981, 23, 2790. McCurdy, C. W.; R e s c i g n o , T. N. Phys. Rev. Letts. 1978, 41, 1364. J u n k e r , B. R. Phys. Rev. Letts. 1980, 44, 1847. J u n k e r , B. R. I n t . J. Quantum Chem. 1980, 14S, 53. Brändas, E.; Froelich, P.; Obcema, C. H.; E l a n d e r , N.; R i t t b y , M. Phys. Rev. A 1982, 26, 3656. McCurdy, C. W.; R e s c i g n o , T. N.; D a v i d s o n , E . R.; L a u d e r d a l e , J. G. J. Chem. Phys. 1980, 73, 3268. M i s h r a , M.; Ohm, Y.; Froelich, P. Phys. Lett. A 1981, 84, 4. R e s c i g n o , T. N.; O r e l , A. E.; McCurdy, C. W. J . Chem. Phys. 1980, 73, 6347. McCurdy, C. W.; L a u d e r d a l e , J. G.; Mowrey, R. C. J. Chem. Phys. 1981, 75, 1835. McCurdy, C. W.; Mowrey, R. C. Phys. Rev. A 1982, 25, 2529. L a u d e r d a l e , J. G.; McCurdy, C. W.; H a z i , A. U. J. Chem. Phys. 1983, 79, 2200. McNutt, J. F.; McCurdy, C. W. Phys. Rev. A 1983, 27, 132. R e s c i g n o , T. N.; McCurdy, C. W.; O r e l , A. E . Phys. Rev. A 1978, 17, 1931. S c h n e i d e r , B. I . ; Le Dourneuf, M.; Vo Ky Lan Phys. Rev. Lett. 1979, 43, 1926. H a z i , A. U.; R e s c i g n o , T. N.; Kurilla, M. Phys. Rev. A 1981, 23, 1089. L e v i n , D. A.; McKoy, B. V. ( u n p u b l i s h e d results). Chung, K. J.; D a v i s , B. F. Phys. Rev. A, 1982, 26, 3278. H i c k s , P. J.; Comer, J. J. Phys. B 1975, 8, 1866. H a z i , A. U.; T a y l o r , H. S. Phys. Rev. A 1970, 1, 1109 g i v e s a detailed treatment o f the stabilization phenomenon. Thompson, T. C.; and T r u h l a r , D. G. Chem. Phys. L e t t s . 1982, 92, 71. McCurdy, C. W.; McNutt, J. F. Chem. Phys. Letts. 1983, 94, 306.

Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

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Simons, J . J . Chem. Phys. 1981, 75, 2465. Mowrey, R. C.; McCurdy, C. W. (unpublished r e s u l t s ) . Bai, Y. Y.; Hose, G.; McCurdy, C. W.; Taylor, H. S. Chem. Phys. Letts. 1983, 99, 342. Froelich, R.; Davidson, E.R.; Brändas, E. Phys. Rev. A 1983, 28, 2641. Golden, D.E.; Schowengerdt, F.D.; Macek, J . J . Phys. B 1974, 7, 478. The resonance energy of reference 20 i s compared with this measurement using a ground state energy of -79.0016 eV and a conversion factor of 27.211652 eV/E . Moiseyev, N.; Froelich, P.; Watkins, E. J . Chem. Phys. 1984, 80, 3623. h

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RECEIVED

June 11, 1984

Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.