Continuum orbitals, complex scaling problem, and the extended virial theorem

Brändas, Erkki; Froelich, Piotr

doi:10.1103/physreva.16.2207

Cited by 167 publications

(52 citation statements)

References 21 publications

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“…Solutions of condition [Eq. (S)] are closely related to the complex virial theorem for scattering resonances, as noted by Brandas et al [9,14], Winkler [ lo], and the present authors [4]. Indeed, from the complex Hellmann-Feynman theorem [4] converged successfully on the second lowest ' S resonance, even when the procedure was started from the correct aopr, Oopt obtained by the Powell method described below.…”

Section: Virial Stationary Pointssupporting

confidence: 51%

Complex‐coordinate studies of helium autoionizing resonances

Moiseyev

Certain

Weinhold

1978

Int J of Quantum Chemistry

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Several aspects of the numerical application of the complex-coordinate method for calculating helium autoionizing resonances are discussed. A direct-search algorithm is used to locate resonance eigensolutions that satisfy the virial theorem, and the influence of the basis set o n the accuracy of the calculated positions and widths is investigated. The complex-coordinate method is related to the stabilization method

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Section: Virial Stationary Pointssupporting

confidence: 51%

Complex‐coordinate studies of helium autoionizing resonances

Moiseyev

Certain

Weinhold

1978

Int J of Quantum Chemistry

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“…near converged results. As has been shown recently by several authors [14], for complex scaling in general, and applied to the Stark problem by Brandas and Froelich [4], these stationary points may be found directly by enforcement of a virial condition, and a subsequent nonlinear variation to determine 8, with overall scaling of the real parts of the coordinates conveniently included by allowing 0 itself to take on complex values. In any case, the "kink" point of Figure 3 suggests a value of 8 = 0.4 (rad) and Table I (from Ref. 2 ) documents that good convergence is obtained in this region.…”

Section: A H Atom In a DC Fieldmentioning

confidence: 97%

Complex coordinates and the Stark effect

Cerjan

Hedges

Holt

et al. 1978

Int J of Quantum Chemistry

106

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The theories of the dilatation, r + r eie, and translation, x --t x + iq, transformations as related to the Stark problem are reviewed, and new results obtained. Results for the hydrogen atom n = 1 and n = 2 levels and the 'Po, 2s2p H-shape resonance in dc fields are presented, and the extension to the ac Stark effect made. Spectral estimates are made using the technique of the numerical range and via discussion of several model problems, using both coordinate rotation and coordinate translation.

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“…(2) Stability of the energy versus 8 guarantees that the righthand side of Eq. (2) is a function of r exp(i8) [13,19]. The Gamow-Siegert definition [1,2] of a resonance allows for the calculation of a quantized complex energy and a complex wave function: The Schrodinger equation is solved with outgoing boundary conditions in all channels open for dissociation of the system.…”

mentioning

confidence: 99%

Back rotation of complex rotated resonance wave functions: A numerical study

Lefèbvre

1992

Phys. Rev. A

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Pade approximants are used to derive the Gamow-Siegert wave function by back rotation of its complex rotated forms. This provides an unambiguous way to normalize a resonance wave function. 2 X (r)= 1l n m. r sin L The complex rotated resonance wave function is written as P", (r exp(i8))= g C"(8)y"(r) .(2) Stability of the energy versus 8 guarantees that the righthand side of Eq. (2) is a function of r exp(i8) [13,19]. The Gamow-Siegert definition [1, 2] of a resonance allows for the calculation of a quantized complex energy and a complex wave function: The Schrodinger equation is solved with outgoing boundary conditions in all channels open for dissociation of the system. Since this leads to complex wave numbers, a component P(r) of the wave function behaving as A exp[ikr] with k =ko ik & -(ko k]positive) becomes divergent as r, the dissociative coordinate, becomes infinite. Much attention has been paid to this circumstance [3 -10] with the aims of (1) offering a normalization method for the wave function (essentially, how to choose the amplitude A ) and (2) allowing the use of Gamow-Siegert wave functions as a basis set, just as is done for normalizable bound-state wave functions. Most of these problems are bypassed by the now very popular complex rotation method [11,12]: Changing r into p=r exp[i8] with 8)tan '(k&lko) leads to a localized wave function that can be normalized to unity [with P (r), however, instead of~P(r)~in the normalization integral [13]]. Since the complex rotated wave function is defined as the analytically continued Gamow-Siegert resonance wave function upon changing r into a complex coordinate, the reverse problem has also been given some attention. Changing p into p exp( i 8) leads back to the original coordinate r. This operation has been called back rotation [14,15]. One use of it is to ease the calculation of matrix elements in basis set approaches to resonance determination [16). When operating on a complex rotated wave function, back rotation should produce the resonance wave function, with the advantage that a unique function is obtained in this way. Several attempts to obtain this function have been described in the literature [15,17,18]. The simplest procedure for a singlechannel case consists in using a basis of integrable functions, such as free-particle wave functions in a box of length L:Back rotation is then obtained by writing P", (r) = g C"(8)y"(rexp( i 8-)) .( 3) However, the sine functions of Eq. (1) become combinations of hyperbolic functions with the result that it is difficult for large r to get satisfactory interference between these very oscillatory basis functions. The use of other basis sets meets with similar difficulties [16,17].We describe in this Brief Report a different approach which is made possible by the accurate determination of complex rotated resonance wave functions through numerical solution of coupled-channel differential equations with matching of functions obeying shortand long-range boundary conditions. This is based on the Fox-Goodwin procedure [...

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Continuum orbitals, complex scaling problem, and the extended virial theorem

Cited by 167 publications

References 21 publications

Complex‐coordinate studies of helium autoionizing resonances

Complex‐coordinate studies of helium autoionizing resonances

Complex coordinates and the Stark effect

Back rotation of complex rotated resonance wave functions: A numerical study

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