Supercritical Dirac resonance parameters from extrapolated analytic continuation methods
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Supercritical Dirac resonance parameters from extrapolatedanalytic continuation methods
Edward Ackad and Marko Horbatsch
Department of Physics and Astronomy, York University,4700 Keele St, Toronto, Ontario, Canada M3J 1P3
Abstract
The analytic continuation methods of complex scaling (CS), smooth exterior scaling (SES), andcomplex absorbing potential (CAP) are investigated for the supercritical quasimolecular groundstate in the U -Cf system at an internuclear separation of R = 20 fm. Pad´e approximantsto the complex-energy trajectories are used to perform an extrapolation of the resonance energies,which, thus, become independent of the respective stabilization parameter. Within the monopoleapproximation to the two-center potential is demonstrated that the extrapolated results from SESand CAP are consistent to a high degree of accuracy. Extrapolated CAP calculations are extendedto include dipole and quadrupole terms of the potential for a large range of internuclear separations R . These terms cause a broadening of the widths at the / level when the nuclei are almost incontact, and at the % level for R values where the 1S σ state enters the negative continuum. . INTRODUCTION The ability to produce super-strong electric fields in experiments, either by super-intenselasers or in collisions with highly ionized heavy atoms, opens the possibility to study nonlin-ear phenomena. It allows for detailed testing of the interface of relativistic quantum mechan-ics and quantum electrodynamics by probing electron dynamics at an energy scale wherephenomena such as electron-positron pair creation may be detected against the backgroundof more conventional atomic physics processes such as ionization and positron productionby time-dependent fields [1].An electron in a Coulomb potential represents a problem that has been studied in bothrelativistic and non-relativistic physics. As the potential depth increases with nuclear charge Z , the ground state’s energy decreases as Z . Since the spectrum of the Dirac equation fora Coulomb potential is not bounded from below (due to the negative-energy continuum E < − m e c ), a potential of sufficient strength can yield resonance states when the ground-state energy E < − m e c . This causes the QED vacuum to become unstable and decayby pair-creation to a charged vacuum state [2]. The potential is called super-critical in thiscase.For a Coulomb potential, a single nucleus would need to be much more massive than anyobserved element ( Z ∼ >
169 for finite-size nuclei), but when two large nuclei get close enough,the combined two-center potential can become supercritical [2, 3]. As the two nuclei getcloser the energy of the quasi-molecular ground state (1S σ ) decreases and so the resonanceenergy is embedded deeper into the negative-energy continuum of the Dirac spectrum. Thesenegative-energy electron states can be reinterpreted using CPT symmetry as positron statesof positive energy. Therefore, by studying these supercritical resonances, the dynamics of2ystems at the energy scale of particle creation can be explored.It is possible to build a matrix representation of the hermitian two-center Hamiltonian fora supercritical potential in order to measure the resonance parameters from the Lorentziandistributions in the energy spectrum or from the density of states [4]. More accurate resultswere obtained from analytic continuation methods which yield discrete resonance states withthe following properties: (i) bounded, square-integrable eigenfunctions,i.e., elements of theHilbert space of the original Hamiltonian; (ii) complex eigenenergies, E R , whose real andimaginary parts agree with the center and width of the Lorentzian shape in the spectrumof the original Hamiltonian. Note that the imaginary part is positive, i.e. , E R = E res + i Γ / E res < − m e c [4].Analytic continuation makes use of some parameter ζ to turn the original Hamiltonianinto a non-hermitian operator. This raises two problems: (i) one has to optimize ζ to findthe best approximation to the complex resonance eigenenergy; (ii) one worries about theeffect of the unphysical ζ parameter on the final result. The optimization in step (i) isperformed by stabilizing some measure that depends on the eigenenergy as a function of ζ (such as, e.g., the magnitude | E R ( ζ ) | ).Recent work for the method of adding a complex absorbing potential (CAP) has demon-strated how to obtain more accurate results by extrapolating the complex E R ( ζ )-trajectoryto ζ = 0 [5]. The present work extends this idea to the relativistic Dirac equation using thecomplementary methods of smooth exterior scaling (SES) and CAP. The stability and pa-rameter independence of the results is demonstrated. The extrapolation technique enablesus to perform an extension of the calculations beyond the monopole approximation to thetwo-center potential. Results are given for the three coupled channels, κ = − , , −
2, fora large range of internuclear separation, i.e., the resonance width broadening due to dipole3nd quadrupole interactions is calculated.
II. THEORYA. Complex absorbing potential method
The method of adding a complex absorbing potential (CAP) to the Hamiltonian hasbeen used extensively in atomic and molecular physics [6, 7, 8, 9], and was put on firmmathematical grounds by Riss and Meyer [10]. It was recently extended to the relativisticDirac equation for Stark resonances [11], and for supercritical resonances [4]. It works byextending the physical Hamiltonian by an imaginary potential which makes the Hamiltoniannon-hermitian. In the case of the Dirac Hamiltonian a CAP is added as a scalar giving,ˆ H CAP = ˆ H − iη ˆ βW ( r ) , (1)where η is a small non-negative parameter determining the strength of the CAP and ˆ β isthe standard Dirac matrix [3]. The function W ( r ) determines the shape of the potentialand is tailored for the specific problem. Currently, the most common use of a CAP is asa stabilization method: one solves the system on an equally spaced mesh of values of η and takes the minimum of (cid:12)(cid:12)(cid:12) η dE R dη (cid:12)(cid:12)(cid:12) η = η opt as the closest approximation to the true resonanceparameters [10].In the present work we chose W ( r ) = Θ( r − r c ) ( r − r c ) (2)for the CAP, where Θ is the Heaviside function. The CAP parameter r c allows for theturn-on of the potential outside of the “bound” part of the wavefunction [12].4 . Smooth exterior scaling method Smooth exterior scaling (SES) is an extension of the complex scaling (CS) method whosemathematical justification was developed, e.g., by W.P. Reinhardt [13], and Moiseyev [14].CS introduces an analytical continuation of the Hamiltonian by scaling the reaction co-ordinate, r , by r → re iθ , and has been used extensively in atomic and molecular physics[15, 16, 17, 18]. CS was recently extended to the relativistic Dirac equation for supercriticalresonances [4] and in 3D for a Coulomb potential with other short range potentials [19]. SESrelies on the same justification as CS, but uses a general path in the complex plane that iscontinuous; when using non-continuous paths one refers to the method as exterior complexscaling (ECS) [14]. A simple path is obtained by rotating the reaction coordinate into thecomplex plane about some finite position, r s , instead of the origin. The transformation thenhas the form, r → r for r < r s ( r − r s ) e iθ + r s for r s ≤ r. (3)It offers the advantage of turning on the scaling at a distance r s which can be chosenappropriately for a given potential shape. In analogy to the r c parameter of the CAPmethod it is natural to choose r s such that the “bound” part of the resonance state is notaffected directly by the complex scaling.This additional freedom introduces some complications not found in CS: unlike CS, SESdoes not always have a minimum in the (cid:12)(cid:12) dE R dθ (cid:12)(cid:12) curve as a function of θ , which makes it difficultto determine a stabilized θ value, θ opt , for E R ( θ ). An approximation can always be madeby finding the cusp of the trajectory in { E res , Γ } space [20], although this does not allowfor a very precise determination of θ opt compared with a minimum in (cid:12)(cid:12) dE R dθ (cid:12)(cid:12) . Alternatively,the parameters can be determined from either the dE res dθ or d Γ dθ curves, since in practice it5s usually found that at least one of them will have a minimum, with the best results foreach parameter coming from its own derivative minimum [21]. Optimal results are obtainedwhen the three derivatives have the same value of θ opt yielding the same values for E res andΓ(1S σ ).Although CAP and SES appear as separate methods, they are not independent. It hasbeen shown how a CAP can be transformed into CS [22] and that SES is related to CAP[23]. It has also been shown that a transformative CAP (TCAP) gives the same Hamiltonianas SES with the unscaled potential and a (small) correction term [24]. C. Pad´e approximant and extrapolation
The goal of the analytic continuation methods is to make the resonance wavefunction, ψ res , a bounded function by choosing an analytic continuation parameter ζ > ζ crit . Although ψ res is an eigenfunction of the physical, hermitian, Hamiltonian, it is not in the Hilbert spacesince it is exponentially divergent. For a sufficiently large critical value of ζ = ζ crit (called θ crit for CS, and η crit for CAP), ψ res becomes a bounded function and is therefore in theHilbert space of the physical Hamiltonian [14]. Taking the lim ζ → E R ( ζ ) always yields a realeigenvalue corresponding to ˆ H ( ζ = 0)Ψ = E R ( ζ = 0)Ψ since ˆ H ( ζ = 0) is hermitian whenacting on bounded functions.The authors of Ref. [5] proposed the following: instead of taking directly the ζ = 0 limit,an extrapolation of a part of the trajectory, E R ( ζ ), namely for ζ > ζ crit , is used to obtainthe complex-valued E Pad´e ( ζ = 0). The points used for the extrapolation are computedeigenvalues restricted to a region where ψ res remains in the Hilbert space (as represented by6he finite basis). For the extrapolation the Pad´e approximant is given by, E Pad´e ( ζ ) = P N i =0 p i ζ i P N +1 j =1 q j ζ j (4)where p i and q j are complex coefficients, and N p = 2( N + 1) is the number of points usedin the approximant [25]. The extrapolated value of E Pad´e ( ζ = 0) is given simply by p , andfollows from a given set of ζ -trajectory points, ǫ i = { ε j = E R ( ζ i + j ∆ ζ ) , j = 1 ...N p } , (5)where ζ i ≥ ζ crit . For practical considerations about the added complications of the method(choice of ǫ i , order of the extrapolation) we refer the reader to section III C. III. RESULTS
To examine the smooth exterior scaling (SES) method and compare it to the simplercomplex scaling (CS) method, and to the method of a complex absorbing potential (CAP),we look at the supercritical system of a single electron exposed to the field of a uranium ( A =238) nucleus separated by 20 fm from a californium ( A = 251) nucleus. The same system wasexplored in Ref. [4] using the CS and CAP methods and in Ref. [1] using phase-shift analysisand numerical integration. The nuclei are approximated as displaced homogeneously chargedspheres with a separation of R between the charge centers. In the center-of-mass frame themonopole potential for each of the two nuclei with respective charge Z i , radius R ( i )n andcenter-of-mass displacement R CM is given (in units of ~ = c = m e = 1 , Z = Z i , R n = R ( i )n )7y V ( r ) = − Zαr for r > r + − ZαR R CM h ( R CM − R n ) ( R CM +3 R n )16 r − r ( R CM − R n )4 + R − R ) r − R CM r + r i for r − < r < r + − ZαR CM for r < r − (6)where r ± = R CM ± R n [26]. The expression is obtained from the potential for a homoge-neously charged sphere displaced by R CM along the z -axis and then expanded in Legendrepolynomials.To solve the Dirac equation for the potential given by Eq. (6), a matrix representation isconstructed using the mapped Fourier grid method [27]. The radial coordinate is mappedto a new variable φ by, r ( φ ) = sφ − sφ ( π − φ ) , (7)were s is a scaling parameter which allows for the tailoring of the N mesh points. Thetransformation maps r ∈ [0 , ∞ ) to φ ∈ [0 , π ) which allows for a more efficient coverage ofthe relevant phase space. A. Pad´e extrapolation of SES trajectories
In SES (as in the other methods) the energy E R ( θ ) forms a trajectory which dependson the calculation parameters ( N, s, r s ). Figure 1 shows the θ -trajectory for N = 250, s = 400, ∆ θ = 0 .
01 and r s = 2 and r s = 3 respectively as a sequence of squares and circlesrespectively. The values are displayed as the deviation from a reference value given by E res = − . e c and Γ = 4 . N = 3000, s = 400 and r c = 2. The uncertainties were determinedfrom calculations with different basis parameters, s .8 ∆ Γ ( k e V ) ∆ E res (10 -5 m e c ) Figure 1: (Color Online) The Γ(1S σ ) resonance θ -trajectory from N = 250, r s = 2 (red squares)and r s = 3 (blue circles) displayed as the difference from a reference calculation (cf. text) forthe U-Cf system at R = 20 fm in monopole approximation. The × ’s mark the resonance energyobtained by minimizing (cid:12)(cid:12)(cid:12) dE R dθ (cid:12)(cid:12)(cid:12) . The N p = 8 data points used for the extrapolations are bracketedby the solid symbols, and a spacing of ∆ θ = 0 .
01 was used.
In Fig. 1, the E R ( θ ) results for small θ values start in the upper right hand and abruptlychange direction after a few points. This happens in the vicinity of the reference value givenby the point (0,0). As θ increases beyond the cusp in the trajectory, E R ( θ ) changes moreslowly, and the θ -trajectory points become denser. Due to the finite number of collocationpoints N , the actual value of θ crit (the minimum θ value such that ψ res is bounded andproperly represented within the finite basis) is higher than theoretically predicted [14], andis close to the cusp-like behavior in the θ -trajectory.Optimal, directly calculated, approximate values for the resonance energy are chosen byfinding the minimum of (cid:12)(cid:12) dE R dθ (cid:12)(cid:12) θ = θ opt [13, 14], and are displayed as black crosses for both9hoices of r s . The solid lines represent N p = 8 Pad´e extrapolation curves (cf. Eq. (4)). Thevalues of ε and ε N p from Eq. (5), i.e., the bracketing points used to determine the Pad´eapproximant, are indicated by filled symbols. The values of θ = 0 .
28 for the r s = 2 curveand θ = 0 .
36 for the r s = 3 curve were chosen, because their E Pad´e ( ζ = 0) points werefound to be closest to each other. Using other starting values for the Pad´e curves wouldresult in indistinguishable extrapolation curves on the scale of Fig. 1 (as long as the startingpoint would not be too close to the cusp in the θ -trajectory). From the Pad´e curves oneobtains the best approximation to the resonance parameters by selecting the end points( E Pad´e ( ζ = 0)), located close to the the origin (0,0), i.e., close to the N = 3000 referencecalculation result.Although very different trajectories from those shown in Fig. 1 were obtained for different r s values, the Pad´e curves still extrapolated to nearly the same ζ = 0 resonance energy, aslong as r s was chosen outside the “bound” part of the wavefunction and not too large, i.e.1 . ∼ < r s ∼ <
6. Larger values of r s pose difficulties for the finite- N computation as it cannotspan the large- r region properly.Simple complex scaling (CS) is equivalent to SES with r s = 0 and, thus, falls outside ofthe range of acceptable r s values. It yields results with a much larger deviation from thereference result, even when Pad´e extrapolation is applied.The scaling parameter, s , introduced by the mapping of the radial coordinate in themapped Fourier grid method in Eq. (7), plays a dominant role as far as numerical accuracyis concerned. In Fig. 2 the magnitude of the relative error in the width Γ(1S σ ) is shown as afunction of s for SES calculations with N = 500, while using two reference CAP calculations.The latter were obtained from the stabilization method with a large basis ( N = 3000) andyielded values of E res = − . . s = 400, and10 res = − . . s = 750 respectively. Each SES data point was -7 -6 -5 -4 -3 -2
0 500 1000 1500 2000 | ∆ Γ | / Γ s Figure 2: (Color Online) Magnitude of relative error for Pad´e extrapolated widths ∆Γ(1S σ ) for N = 500 as a function of the scaling parameter, s , for the U-Cf system at R = 20 fm in monopoleapproximation. Data shown are obtained using two reference calculations from the stabilized CAPmethod with N = 3000 and r c = 2; red plus symbols: s = 400; blue squares: s = 750 (cf. text). obtained by finding the closest intersection of the r s = 2 , , θ ’s (for θ > θ crit ), and taking the average of these three best values. The use of different r s values within the acceptable range would cause changes too small to see on this plot. Theresonance position, E res , is much more stable with respect to s and therefore not shown.The technique of combining information from different r s calculations in order to selectthe ideal subset of θ points helps with the following problem. For each calculation withfixed value of r s we looked at extrapolated values for the resonance width as a function ofthe start value θ i , and observed the deviation from the average value. It was found thatnumerical noise present in the data was minimal for some range of θ i values (different for each11alculation). This noise was investigated as a function of extrapolation order N p . Combininginformation from calculations with different values of r s allowed us to eliminate two sourcesof random uncertainties associated with finite-precision input to a sensitive extrapolationcalculation.The results demonstrate that the extrapolated ( θ = 0) SES N = 500 calculations exceedthe precision of the stabilized (non-extrapolated) N = 3000 CAP calculations. Table I givesthe resonance parameter results ( E res , Γ) for two different extrapolations, namely N p = 4 , N = 500, and mapping parameter 300 ≤ s ≤ σ = N i X i p ( x i − ¯ x ) N i , (8)where ¯ x is the average x -value, are given separately for resonance position and width. Theresults are encouraging since extrapolation provides a marked improvement over the stabi-lized CS method discussed in Ref. [4]. N p E res σ ( E res ) Γ (keV) σ (Γ)8 -1.75793073 1.7 × − × − × − × − Table I: Averaged resonance position and width from the extrapolated SES method with basis size N = 500 using extrapolation orders N p = 8 and N p = 4. The values were averaged over the stablerange 300 ≤ s ≤ σ separately for positionand width. . Pad´e extrapolation of CAP trajectories We have shown that a quadratic complex absorbing potential (cf. Eq. 2) gives betterresults than simple complex scaling (CS) when used as a stabilization method [4]. We findthat the performance of SES represents an improvement over that of CS and it is competitivewith CAP.Even for the same calculation parameters (
N, s, r c ), the η -trajectory for CAP is foundto be different from the θ trajectory in SES. Figure 3 shows a trajectory pair for identicalcalculation parameters, namely N = 500, s = 600, and r s = r c = 3 respectively. Even -0.002-0.001 0 0.001 0.002 0.003 0.004 -1.5 -1.0 -0.5 0 0.5 ∆ Γ ( k e V ) ∆ E res (10 -6 m e c ) Figure 3: (Color Online) SES θ -trajectory with N = 500, s = 600, r s = 3 (blue circles), andCAP η -trajectory for N = 500, s = 600, r c = 3 (red diamonds) displayed as the difference from areference calculation for the 1S σ state in the U-Cf system at R = 20 fm in monopole approximation.The × ’s mark the approximate values obtained from the stabilization method. The reference valuewas obtained by Pad´e extrapolation of CAP calculations with N = 3000, N p = 4 for r c = 2 , , E res = − . e c and Γ = 4 . E Pad´e ( η = 0)) as ( E Pad´e ( θ = 0)) from the SES method, i.e. the extrapolated resonanceparameters for both methods are very similar. When looking at plots of the N = 500 η = 0extrapolation results for CAP as a function of s , as was done in Fig. 2 for SES, we observealmost identical results. In table II, the average and standard deviation of the mean, σ (cf. Eq. 8), for the CAP results for N = 500, 300 ≤ s ≤
700 are given for two differentextrapolation sizes, N p = 4 , N p E res σ E res Γ (keV) σ Γ × − × − × − × − Table II: Same as in Table I, but for the extrapolated CAP method with basis size N = 500. C. Comparison of CAP and SES results
Both methods (CAP and SES) show similar behavior with respect to different aspects ofthe calculation. Both are relatively insensitive to the number of Pad´e points, N p , used inthe extrapolation. It was found that the results were unchanged, to the relevant precision,for 4 ≤ N p ≤
12. The effects of changing the distance between the analytic continuationparameter (∆ θ for SES and ∆ η for CAP) were similarly found to be small compared with s . It was found that one could choose optimal ζ i start values for the Pad´e approximationby minimizing the deviation between E R ( ζ = 0) from different r s or r c calculations. This issimple to implement, and works well for different values of N p . In this way one obtains resultsthat are independent of the starting point for the rotation ( r s in SES) or the imaginary14otential ( r c in CAP). The results are therefore very stable with respect to the analyticcontinuation parameters and depend only on the parameters from the mapped Fourier gridmethod. As shown in Fig. 2 for SES (a very similar graph was obtained for CAP), there isa range of s for which results are stable. For larger basis size N the stable s -range increasesmaking a judicious choice of s less important.We have averaged the results from both the SES and CAP methods over the stable s region, for a basis size of N = 500 in tables I & II. The standard deviation for the width σ (Γ)within either method is below 10 − , while the results differ at this level. It is, therefore, ofinterest to determine the reliability of the error estimate which is based upon basis parametervariations using larger- N extrapolated calculations. Comparison of the width results withsuch an estimate based upon N = 3000 CAP and SES calculations indicates that SES andCAP converge to the same value, closest to the CAP value given in table II.Concerning the most appropriate order for the extrapolations it is worth noting that theCAP Pad´e trajectories are rather straight in comparison with the ones for SES, and N p = 4might be more appropriate in this case. Nevertheless, we find no systematic improvementwhen going to the lower-order approximation (which might be deemed more stable withrespect to numerical noise in the trajectory points). IV. COUPLED-CHANNEL CALCULATIONS
The ability to compute the resonance parameters to high precision with moderate basissize (e.g., N = 500) allows us to explore the effects of higher multipoles which are presentin the two-center interaction. We are not aware of prior investigations of such coupled-channel resonance calculations of supercritical Dirac states. While the effect on the resonanceposition is expected to be small, the sensitivity of the width to computational details (cf.15he different results discussed in [4]) indicates that some broadening of the resonance mayoccur.To account for such two-center potential effects the wavefunction is expanded using spinorspherical harmonics, χ κ,µ ,Ψ µ ( r, θ, φ ) = X κ G κ ( r ) χ κ,µ ( θ, φ ) iF κ ( r ) χ − κ,µ ( θ, φ ) , (9)which are labeled by the relativistic angular quantum number κ (analogous to l in non-relativistic quantum mechanics) and the magnetic quantum number µ [3]. The Diracequation for the scaled radial functions, f ( r ) = rF ( r ) and g ( r ) = rG ( r ), then becomes( ~ = c = 1), df κ dr − κr f κ = − ( E − g κ + ±∞ X ¯ κ = ± h χ κ,µ | V ( r, R ) | χ ¯ κ,µ i g ¯ κ , (10) dg κ dr + κr g κ = ( E + 1) f κ − ±∞ X ¯ κ = ± h χ − κ,µ | V ( r, R ) | χ − ¯ κ,µ i f ¯ κ , (11)where V ( r, R ) is the potential for two uniformly charged spheres displaced along the z -axis,which is expanded into Legendre polynomials according to V ( r, R ) = P ∞ l =0 V l ( r, R ) P l (cos θ )[3]. The monopole term, V , is given explicitly in Eq. (6), and for the present work we includethe coupling terms required for the κ = ± , − V ( r, R ) h χ ± κ,µ | P | χ ± ¯ κ,µ i dipole and V ( r, R ) h χ ± κ,µ | P | χ ± ¯ κ,µ i quadrupole terms). The κ = 1 , − P / and P / respectively) have the strongest coupling to the κ = − σ resonance parameters for different internuclear separations forboth the one- and three-channel CAP calculations. The CAP calculations were performedusing a basis size of N = 500 per channel with 300 ≤ s ≤ r c = 2 , , θ range for extrapolation, which was carried16 (fm) Single-channel ( κ = −
1) Three-channel ( κ = − , , | Γ − Γ | Γ E res (mc ) Γ (keV) E res (mc ) Γ (keV) ( × − )16 -2.00635363(3) 8.148233(4) -2.00646180(3) 8.150153(3) 0.23618 -1.87487669(4) 5.881605(1) -1.87502343(4) 5.883977(2) 0.40320 -1.75793073(4) 4.0848148(4) -1.75811259(4) 4.087394(1) 0.63122 -1.65393272(3) 2.708987(1) -1.65414448(3) 2.711524(2) 0.93724 -1.5612122(1) 1.6966694(3) -1.56144826(1) 1.698947(1) 1.3426 -1.47820830(4) 0.9877052(4) -1.47846358(4) 0.989571(1) 1.8928 -1.40354329(5) 0.5221621(4) -1.40381341(4) 0.523544(1) 2.6530 -1.33603643(6) 0.2420749(2) -1.33631779(6) 0.242976(1) 3.7232 -1.27469286(5) 0.0931429(1) -1.27498253(5) 0.093639(5) 5.3234 -1.21868219(5) 0.0271161(1) -1.21897787(4) 0.027322(2) 7.5936 -1.16731491(6) 0.0050374(1) -1.16761472(7) 0.005091(2) 10.738 -1.12001745(6) 0.0004170(3) -1.12031983(6) 0.000439(4) 52.5Table III: Averaged 1S σ resonance position and width from the extrapolated CAP method withbasis size N = 500, N p = 4, for single-channel (monopole) and three-channel calculations as afunction of separation, R , in the U-Cf system. The values were averaged over the stable range300 ≤ s ≤
700 using r c = 2 , ,
4. The value in parenthesis represents the standard deviation fromthe average and the last column gives the relative difference of the width from three- to one-channel. out with order N p = 4. Values in parentheses indicate the standard deviation of the mean(cf. Eq. 8). We note that the accuracy of the calculations - as indicated by the deviation -is much higher than what is required to measure the effect of the the P-state channels onthe 1S σ resonance. 17he final column contains the relative difference of the width between the three-channelresults and the monopole approximation. The correction due to the dipole and quadrupolepotentials is seen to increase with internuclear separation R . This trend can be expected,as the overall interaction becomes less spherically symmetric in this limit. The effect issmall, however, as the strongest contribution towards the binding energy (and thus thesupercriticality) is provided by the monopole part. The dipole interaction contributionsto the resonance parameters are a result of the relatively small charge asymmetry in theU -Cf potential. Quadrupole couplings to D states can be expected to generate moresignificant changes in E res and Γ. V. DISCUSSION
We have extended the method of smooth exterior scaling (SES) to the relativistic Diracequation. For both the addition of a complex absorbing potential (CAP) and SES, we havealso shown that by extrapolation of the complex energy eigenvalue trajectories which arefunctions of the analytic continuation parameter, ζ , to ζ = 0 using appropriate ζ -trajectoryvalues (cf. section II C), one obtains an accurate and robust estimation of the resonanceparameters. These results use much smaller basis than would be required for similar pre-cision using stabilization. While adding the extrapolation adds new parameters, we foundminimizing the distance between extrapolated results for different calculation parameterswas a simple and highly effective method of reducing the parameter space. The effects ofthe number of calculation points, N p , was found to be very small provided it was in therange 4 ≤ N p ≤ P -states were explored and found to have a in-creasing influence as the two-center system becomes less spherically symmetric which occurs18t larger internuclear separations R . The resonances could acquire additional broadeningsand shifts from dynamical effects not included in the quasi-static approximation. VI. ACKNOWLEDGMENTS [1] J. Reinhardt, B. M¨uller, and W. Greiner, Phys. Rev. A , 103 (1981).[2] J. Rafelski, L. Fulcher, and A. Klein, Phys. Rep. , 227 (1978).[3] W. Greiner, Relativistic Quantum Mechanics: Wave Equations (Springer, 2000), 3rd ed.[4] E. Ackad and M. Horbatsch, Phys. Rev. A , 022508 (2007).[5] R. Lefebvre, M. Sindelka, and N. Moiseyev, Phys. Rev. A , 052704 (2005).[6] M. Ingr, H.-D. Meyer, and L. S. Cederbaum, J. Phys. B , L547 (1999).[7] R. Santra, R. W. Dunford, and L. Young, Phys. Rev. A , 043403 (2006).[8] R. Santra and L. S. Cederbaum, J. Chem. Phys. , 5511 (2002).[9] S. Sahoo and Y. K. Ho, J. Phys. B , 2195 (2000).[10] U. V. Riss and H.-D. Meyer, J. Phys. B , 4503 (1993).[11] I. A. Ivanov and Y. K. Ho, Phys. Rev. A , 023407 (2004).[12] I. B. M¨uller, R. Santra, and L. S. Cederbaum, Int. J. Quant. Chem. , 75 (2003).[13] W. P. Reinhardt, Ann. Rev. Phys. Chem. , 223 (1982).[14] N. Moiseyev, Phys. Rep. , 211 (1998).
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