Complex coordinate rotation method based on gradient optimization
CComplex coordinate rotation method based on gradient optimization
Zhi-Da Bai,
1, 2
Zhen-Xiang Zhong, ∗ Zong-Chao Yan,
3, 1, 4 and Ting-Yun Shi State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China Department of Physics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3 Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China (Dated: July 24, 2020)In atomic, molecular, and nuclear physics, the method of complex coordinate rotation is a widelyused theoretical tool for studying resonant states. Here, we propose a novel implementation of thismethod based on the gradient optimization (CCR-GO). The main strength of the CCR-GO methodis that it does not require manual adjustment of optimization parameters in the wave function;instead, a mathematically well-defined optimization path can be followed. Our method is provento be very efficient in searching resonant positions and widths over a variety of few-body atomicsystems, and can significantly improve the accuracy of the results. As a special case, the CCR-GO method is equally capable of dealing with bound-state problems with high accuracy, which istraditionally achieved through the usual extreme conditions of energy itself.
PACS numbers: 31.15.-p,34.80.-i,34.85.+x
Resonant states play an important role in atomic,molecular, and nuclear physics and have a long his-tory of research, such as doubly-excited states in two-electron systems [1, 2], Efimov states in weakly boundfew-body systems [3, 4], resonance phenomena under De-bye plasma environment [5], four-body resonant statesin positronium hydride [6] and positron-helium [7], reso-nances in positron scattering by atoms and molecules [8],metastable states in antiprotonic helium ¯p He + [9–11],and resonance phenomena in nuclear physics [12, 13].There exist many theoretical methods for studying res-onant states. In early years, the S -matrix [14] and R -matrix [15] theories were used to solve resonant prob-lems. In 1970s, the method of complex coordinate rota-tion (CCR) was mathematically established [16], and wasfirst used in studies of scattering involving three chargedparticles by Raju and Doolen [17]. After that, the CCRmethod was further developed to calculate atomic reso-nant states by Ho [18]. From then on, the CCR methodhas been widely adopted as a powerful tool for investi-gating resonant states in atoms and molecules, includ-ing its application to high-precision antiprotonic heliumspectrum [10, 11]. On the other hand, Feshbach in1962 [19] formulated a general theory for studying reso-nances, where the wave function space is partitioned intoclosed- and open-channel segments. The hypersphericalclose-coupling method, developed by Lin [20] in 1984 tocalculate doubly-excited states, was applied to positron-atom scattering [21]. Recently, the stabilization methodcombined with hyperspherical coordinates and B -splineexpansion was applied to positron-atom scattering byHan and co-workers [22]. Among these methods, boththe CCR method and the closed-channel approximation ∗ [email protected] of the Feshbach theory can reach high precision for long-lived metastable states with small widths, such as 10 − atomic units in ¯p He + decaying via a radiative chan-nel [9]. However, an Auger-dominated state in ¯p He + is usually short-lived and possesses a width larger than10 − atomic units [9], such as the ( N = 31, L = 30)state, where N and L are, respectively, the principal andtotal angular momentum quantum numbers. The accu-racy of the closed-channel approximation of the Feshbachtheory is limited by the width of a resonant state [23],whereas the accuracy of the CCR method can go beyondthis limit [10], making the CCR method more suitablefor short-lived metastable states.Since the variational approach using Hylleraas- orSturmian-type basis sets has been proven to be effectivein dealing with atomic or molecular few-body systems,it is natural to combine these basis sets with the CCRmethod [24–27], and solve resonance problems variation-ally. However, due to the lack of extreme theorem for aresonance state, historically it is common practice in us-ing the CCR method that the nonlinear variational pa-rameters in the trial wave function are optimized throughrepeated trial and error manual adjustment, which couldbecome extremely laborious and inefficient, especially fora high-dimensional parameter space. In this Letter, wepropose a novel approach of complex coordinate rota-tion based on the gradient optimization (CCR-GO). Theadvantage of the CCR-GO method over the existing res-onance methods is that it does not require manual ad-justment of nonlinear parameters in the wave function;instead, a mathematically well-defined optimization pathcan be followed, leading to a resonance pole quickly. Ourmethod will be tested for various three-body atomic sys-tems.In the method of complex coordinate rotation [18], un-der the radial coordinate transformation r → r exp( iθ ), a r X i v : . [ phy s i c s . a t o m - ph ] J u l the original Hamiltonian of the system ˆ H = ˆ T + ˆ V , whereˆ T and ˆ V are, respectively, the kinetic and potential en-ergy operators, is transformed intoˆ H → ˆ H ( θ ) = ˆ T exp( − iθ ) + ˆ V exp( − iθ ) , (1)where the rotational angle θ is assumed to be real andpositive. According to the Balslev-Combes theorem [16],in the complex energy plane, for sufficiently large θ thistransformation rotates the continuum spectrum of ˆ H to“expose” the resonant poles around the thresholds fromthe unphysical sheet to physical sheet of the Riemannsurface, and the bound state poles remain unchanged onthe negative side of the real axis. The eigenenergies canbe obtained by solving the following complex eigenvalueproblem ˆ H ( θ )Ψ θ = E Ψ θ , (2)where the eigenfunction Ψ θ is square integrable and thecorresponding discrete complex eigenvalue E = E r − i Γ / E r and the width Γ of a resonance.By choosing a basis set { ψ n , n = 1 , . . . , N } in an N -dimensional Hilbert space, the complex eigenvalue prob-lem (2) can be converted to the following generalized al-gebraic complex eigenvalue problem H ( θ )Ψ θ = E O Ψ θ , (3)where H ( θ ) ij = (cid:104) ψ i | ˆ H ( θ ) | ψ j (cid:105) are the N ×N
Hamiltonianmatrix elements and O ij = (cid:104) ψ i | ψ j (cid:105) are the overlap ma-trix elements. Since a resonance wave function is squareintegrable, the rotated Hamiltonian ˆ H ( θ ) holds the com-plex variational principle that makes the complex energyeigenvalue stationary, although not necessarily extreme,with respect to any parameter ξ in the wave function,such as the rotational angle θ , or a nonlinear parameterin a Hylleraas basis set, or the box size of a B -spline basisset, i.e. , ∂ ξ E ≡ ∂E∂ξ = 0 (4)at a resonance pole. This expression can be understoodas a stability condition for a resonant energy, which ofcourse also applies to any bound state as a special case.Since we do not have the extreme theorem for a resonanceenergy E in general, instead of dealing with E itself, wefocus on | ∂ ξ E | and minimize it by varying ξ , due to theobvious fact that | ∂ ξ E | ≥
0. This is the essence of ourCCR-GO method.To be specific, let us consider a three-body Coulombicsystem, such as Ps − , H − , He, and ¯p He + . After eliminat-ing the center of mass coordinates, a three-body problemis reduced to a quasi two-body one with (cid:126)r and (cid:126)r beingtheir position vectors relative to the third particle. In or-der to solve the complex eigenvalue problem (2), we usetwo types of basis sets. The first one consists of Hylleraasfunctions with real nonlinear parameters α and β : { r (cid:96) r m r n e − αr − βr Y LM(cid:96) (cid:96) (ˆ r , ˆ r ) } , (5) (cid:1)(cid:3)(cid:2)(cid:3)(cid:10)(cid:9)(cid:3)(cid:9) (cid:1)(cid:3)(cid:2)(cid:3)(cid:10)(cid:9)(cid:3)(cid:8) (cid:1)(cid:3)(cid:2)(cid:3)(cid:10)(cid:9)(cid:3)(cid:7) (cid:1)(cid:3)(cid:2)(cid:3)(cid:10)(cid:9)(cid:3)(cid:6)(cid:1)(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:4)(cid:3)(cid:1)(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:11)(cid:1)(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:9)(cid:1)(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:7)(cid:1)(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:5) (cid:4) (cid:5) (cid:6)(cid:7)(cid:8)(cid:9)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:9) (cid:3) (cid:4) (cid:5)(cid:5) (cid:2)(cid:1) (cid:2)(cid:2) (cid:4)(cid:5)(cid:1)(cid:3)(cid:2) (cid:11)(cid:12)(cid:2)(cid:10)(cid:3) (cid:1)(cid:13) (cid:1) (cid:2)(cid:10)(cid:3)(cid:1)(cid:13) (cid:2) (cid:2)(cid:10)(cid:3) (cid:2)(cid:5)(cid:2)(cid:1)(cid:2)(cid:7)(cid:6)(cid:2)(cid:5)(cid:2)(cid:8)(cid:6)(cid:4)(cid:1)(cid:5)(cid:4)(cid:1)(cid:3)(cid:8)(cid:9)(cid:10)(cid:5)(cid:7)(cid:3) FIG. 1. Two optimization paths based on (cid:126)g ( E ) and (cid:126)g ( E )for the lowest resonant state S e in Ps − below the Ps ( N = 2)threshold, with the size of basis set N = 252. The inset is anenlarged view of the paths around the convergence point. Inatomic units. (cid:1)(cid:5)(cid:2)(cid:9)(cid:3)(cid:7) (cid:1)(cid:5)(cid:2)(cid:9)(cid:3)(cid:5) (cid:1)(cid:5)(cid:2)(cid:9)(cid:3)(cid:3) (cid:1)(cid:5)(cid:2)(cid:8)(cid:9)(cid:8)(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)(cid:3)(cid:4)(cid:3)(cid:2)(cid:3)(cid:5)(cid:3)(cid:2)(cid:3)(cid:6) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:3)(cid:5)(cid:3)(cid:6)(cid:4)(cid:1)(cid:4)(cid:2) (cid:3)(cid:5)(cid:3)(cid:6)(cid:4)(cid:1)(cid:4)(cid:2) (cid:4)(cid:5)(cid:1)(cid:3)(cid:2) (cid:11)(cid:12)(cid:2)(cid:10)(cid:3)(cid:1)(cid:13) (cid:1) (cid:2)(cid:10)(cid:3)(cid:1)(cid:13) (cid:2) (cid:2)(cid:10)(cid:3) (cid:2)(cid:5)(cid:3)(cid:1)(cid:8)(cid:2)(cid:4)(cid:7)(cid:3)(cid:5)(cid:4)(cid:6)(cid:11)(cid:7)(cid:4)(cid:1)(cid:7)(cid:6)(cid:9)(cid:8)(cid:12)(cid:5)(cid:10)(cid:3) FIG. 2. Two optimization paths based on (cid:126)g ( E ) and (cid:126)g ( E )for the ground state of helium, with the size of basis set N =252. The inset is an enlarged view of the paths around theconvergence point. In atomic units. where Y LM(cid:96) (cid:96) (ˆ r , ˆ r ) is the angular momenta ( (cid:96) , (cid:96) )-coupled spherical harmonics to form a common eigen-state of L and L z . It is noted that a proper sym-metrization of the final wave function is implied for asystem containing two identical particles. The possiblevalues of (cid:96) and (cid:96) are those fulfilling (cid:96) + (cid:96) = L fora state of natural parity ( − L or (cid:96) + (cid:96) = L + 1 fora state of unnatural parity ( − L +1 . Each configuration( (cid:96) , (cid:96) ) has its own set of nonlinear parameters. In orderto enhance the rate of convergence, we may further dividethe most important configuration into more sub-groupseach having different set of nonlinear parameters. Thebasis set is generated by including all terms such that (cid:96) + m + n ≤ Ω with integer Ω controlling the size of basis
TABLE I. Resonance parameters ( E r , Γ /
2) for various three-body Coulombic sys-tems. In the table, N is the size of basis set. A comparison with some of the besttheoretical results is also presented. In atomic units.Author (year) Ref. N − E r Γ / − S e , below Ps ( N = 2) thresholdHo (1979) [28] 161 0.076030(1) 0.000021(1)Li and Shakeshaft (2005) [26] 10206 0.07603044235 0.00002151725This work − S -wave shape resonance, above H ( N = 2) thresholdB¨urgers and Lindroth (2000) [25] 34447 0.103035676 0.015627312Kar and Ho (2012) [27] 700 0.1030357(50) 0.0156273(50)This work S e (1), below H ( N = 2) thresholdHo (1981) [29] 161 0.77787 0.00227Gning et al. (2015) [30] 0.777865 0.002265This work
715 0.7778676356(3) 0.0022706527(1)He P o (1), below H ( N = 2) thresholdHo (1981) [29] 165 0.7604975 0.0001485This work
969 0.76049238762(3) 0.0001494308(1)He D e (1), below H ( N = 2) thresholdHo and Bhatia (1991) [31] 1230 0.7019457 0.0011811This work