Direct Variational Methods for Complex Resonance Energies

2 Direct Variational Methods for Complex Resonance Energies

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 26, 2017 | http://pubs.acs.org Publication Date: September 28, 1984 | doi: 10.1021/bk-1984-0263.ch002

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.

RESONANCES

18


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


( 4 )

2.

MCCURDY

Complex Resonance Energies

19

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.


2.

McCURDY

2. 3. 4. 5. 6.


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

33

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.


34 38. 39. 40. 41. 42.

RESONANCES

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


43.

RECEIVED

June 11, 1984


Direct Variational Methods for Complex Resonance Energies

Recommend Documents