Shape resonances of Be − and Mg − investigated with method of analytic continuation
SShape resonances of Be − and Mg − investigated with method of analyticcontinuation Roman ˇCur´ık ∗ and Ivana Paidarov´a J. Heyrovsk´y Institute of Physical Chemistry,ASCR, Dolejˇskova 3, 18223 Prague, Czech Republic
Jiˇr´ı Hor´aˇcek † Institute of Theoretical Physics, Faculty of Mathematics and Physics,Charles University in Prague, V Holeˇsviˇck´ach 2, 180 00 Prague, Czech Republic (Dated: October 17, 2018)The regularized method of analytic continuation is used to study the low-energy negativeion states of beryllium (configuration 2 s εp P ) and magnesium (configuration 3 s εp P )atoms. The method applies an additional perturbation potential and it requires only routinebound-state multi-electron quantum calculations. Such computations are accessible by mostof the free or commercial quantum chemistry software available for atoms and molecules.The perturbation potential is implemented as a spherical Gaussian function with a fixedwidth. Stability of the analytic continuation technique with respect to the width and withrespect to the input range of electron affinities is studied in detail. The computed resonanceparameters E r =0.282 eV, Γ=0.316 eV for the 2 p state of Be − and E r =0.188 eV, Γ=0.167for the 3 p state of Mg − , agree well with the best results obtained by much more elaboratedand computationally demanding present-day methods. ∗ [email protected] † [email protected]ff.cuni.cz a r X i v : . [ phy s i c s . c h e m - ph ] M a r I. INTRODUCTION
Resonances in electron-atom or electron-molecule scattering, also addressed as transient neg-ative ions, have attracted attention over the last decades. It is because these temporary statesprovide a pathway for electron-driven chemistry via dissociative electron attachment (DEA) andtherefore, applications can be found in chemistry of the planetary atmospheres [1], nanolithogra-phy in microelectronic device fabrication [2, 3], and in cancer research where these states providea mechanism for the DNA damage by low-energy electrons [4, 5].Accurate calculation of energies and lifetimes of the resonances represents a challenging task thatis more complicated than the determination of energies of the bound atomic or molecular states.Temporary negative ions differ from the bound states in two important respects: (i) they arenot stable and decay into various continua, (ii) corresponding poles of the S -matrix are complexand they are expressed by E = E r − i Γ /
2. There have been numerous studies published usingseveral methods for determination of the resonance energies and widths. Stabilization methods[6–9] search for a region of stability of the energies with respect to different confining parameters.Stieltjes imaging technique [8] allows to represent the resonant state by a square-integrable basisand the width is defined by the resonance-continuum coupling. Complex rotation methods [10–12]and the methods employing complex absorbing potential [13, 14] compute complex resonant energyas an eigenvalue of a complex, non-Hermitian Hamiltonian.Recently the method of analytic continuation in coupling constant (ACCC) [15–17] has beenapplied to several molecular targets, such as N [18, 19], ethylene [20, 21], and amino acids [22].Furthermore, the known low-energy analytic structure of the resonance was incorporated intothe inverse ACCC (IACCC) method providing so-called regularized analytic continuation (RAC)method. The RAC method was successfully employed for determination of π ∗ resonances of acety-lene [23] and diacetylene [24] anions, proving that the ACCC method can yield accurate resonanceenergies and widths for various molecular systems using data obtained with standard quantumchemistry codes.Common feature of all methods of analytic continuation is an application of the perturbationpotential λV to the multi-electron Hamiltonian H , i.e. H → H + λV . The role of this attractiveperturbation is to transform the resonant state into a bound state. Although the RAC methodwas developed for strictly short-range perturbation V , authors were able to successfully use theCoulomb potential in its stead [23, 24]. This obvious inconsistency can yield reasonable results,because in practical applications the perturbation potential is often projected on a finite set ofshort-range basis functions, e.g. Gaussian functions used by the quantum chemistry software.However, so obtained weakly-bound states need to be examined carefully because they may, in fact,be Rydberg states supported by the basis and the long-range tail of the Coulomb perturbation V [24]. Such states need to be excluded from the continuation procedure as they do not represent aresonance transferred to a bound state. In order to avoid such complications, in the present studywe adopt a short-range perturbation potential in a form the Gaussian function V ( r ) = − λe − αr . (1)This choice of the perturbation was recently evaluated by White et al. [19] and applied to thewell-known Π g resonance of N − . Furthermore, Sommerfeld and Ehara [25] introduced anothershort-range potential, termed as Voronoi soft-core potential, which they successfully used to analyzethe Π u resonance of CO − .Present analysis of the Gaussian perturbation potential (1) will be carried out for expectedlysimpler problems - atomic shape resonances of beryllium and magnesium. Both atoms are known topossess a p -wave shape resonance very close to the elastic threshold. While in the case of the Mg − the agreement between the available computed resonance parameters [26–28] and the experimentaldata [29, 30] is quite good, the situation is very different for the beryllium atom. There has beena great number of theoretical studies [31–42] aiming to numerically characterize Be − s εp P resonance, with various levels of success. Table III in Ref. [41] clearly summarizes that the theoryof the last four decades predicts the resonance position between 0.1 and 1.2 eV and the resonancewidth between 0.1 and 1.7 eV. Even the most recent calculations differ by about a factor of 3for the two resonant parameters. Moreover, there are no experimental data available for the Be − resonance that could narrow the spread of all the available theoretical predictions.Convergence patters shown in Refs. [41, 42] demonstrate that the Be − resonance may be verysensitive to an accurate description of the electronic correlation energy. Therefore, in the presentstudy we employ coupled-clusters (CCSD-T) and full configuration interaction (FCI) methods forthe perturbed Be − electron affinities that will be then continued the complex plane by the RACmethod. The basic ideas of the RAC method are given in the Sec. II. Quick summary of thequantum chemistry details is presented in Sec. III. In Sec. IV we analyze the stability and accuracyof the RAC method with the Gaussian perturbation potential (1). Then conclusions follow. II. RAC METHOD
The RAC method represents a very simple method for calculation of resonance energies andwidths which embraces all known analytical features of coupling constant λ ( κ ) near the zero energy[24]. The method works as follows: • The atom or molecule is perturbed by an attractive interaction V multiplied by a realconstant λ H neutral → H neutral + λ V , (2)and bound states energies E Ni of the neutral state are calculated for a set of values λ i . • The same procedure is carried out for the corresponding negative ionH ion → H ion + λ V , (3)where the bound state energies E Ii are calculated for the same values of λ i . • Both energies are subtracted forming the electron affinity in the presence of the perturbationpotential
V E Ni − E Ii = E i = κ i . (4)The new set of data points { κ i , λ i } is then used to fit the function λ ( κ ) = λ ( κ + 2 α κ + α + β )(1 + δ κ ) α + β + κ (2 α + δ ( α + β )) . (5)It is represented as a Pad´e 3/1 function and it defines the level of complexity of the pole behaviorat the low bound or continuum energies. We term it as RAC [3/1] method. The origin of its formand the fit formulae for [2/1], [3/2], and [4/2] methods can be found in Ref. [23]. The parametersof the [3/1] fit, namely α, β, δ and λ are found by minimizing the χ functional χ = 1 N N (cid:88) i =1 ε i | λ ( κ i ) − λ i | , (6)where N denotes the number of the points used, while κ i and λ i are the input data. Once anaccurate fit is found, only the parameters α and β determine the resonance energy E r = β − α , (7)and the resonance width Γ = 4 βα . (8)Role of the parameter δ is to describe a virtual state with E v = − /δ . Even in the case thestudied system does not possess a virtual state this parameter represents a cumulative effect of theother resonances and other poles not explicitly included in the model. The weights ε i (accuracyof the data) in Eq. (6) are generally unknown. The calculation can be routinely performed withconstant ε i = 1 or, if an importance of the data points closest to the origin needs to be stressed,increasing weights sequence (e.g. ε i = i) can be used.The RAC method has been recently critically evaluated by White et al. [19]. Authors testedthree types of the perturbation potential V ( r ) = − λr , (9) V ( r ) = − λ e − αr r , (10) V ( r ) = − λe − αr , (11)and they suggested that the attenuated Coulomb potential (10) is the best choice out of the threeoptions and the Gaussian potential (11) does not represent a good choice for the RAC method.All these potentials are easily implemented into the standard quantum chemistry codes. The aimof the present contribution is twofold: • to explore application of the Gaussian-type perturbation and to find its parameters thatallow accurate extraction of the resonance data with the RAC method • to demonstrate that the RAC method can be applied with success to low-lying atomic shaperesonancesBefore applying the RAC method one must consider two important issues.1. First is a choice of the perturbation potential, i.e. in the present context the choice of theexponent α in Eq. (11). Presently there exist no general rule, no guide that helps us tochoose the perturbation potential. Therefore, it is necessary to perform calculations for aset of values of the parameter α to find an optimal choice. If the optimal range of values isfound, it is reasonable to expect that the obtained resonance data should stabilize in sucha range, because the exact function λ ( κ ) gives the same resonant data for every choice ofthe perturbation potential. Since the present [3/1] RAC function is only approximative, onecan only expect an existence of a plateau that gives approximative values of the resonanceparameters.2. The RAC method represents essentially a low-energy approximation to the exact function λ ( κ ). It is therefore obvious that the method should be used in a range of energies (ormomenta) limited by some maximal energy E M . Our empirical experience shows that E m ∼ E r ( E r is the sought resonance energy) gives a reasonable estimate for the range of energies. III. ELECTRON AFFINITIES
Ab initio calculations for the electron affinities E i ( λ i ) in presence of the external Gaussian field(11) were carried out using the CCSD-T [43, 44] and FCI methods as implemented in MOLPRO 10package of quantum-chemistry programs [45]. Core of the basis set employs Dunning’s augmentedcorrelation-consistent basis of quadruple-zeta quality aug-cc-pVQZ [46] for both atoms, Be andMg. This basis set was additionally extended, in an even-tempered fashion, by 2 ( s , d , f , g )-type functions and 6 p -type functions. Calculations for the neutral atoms and corresponding Scaling parameter λ (a.u.) E l ec t r on a ff i n it y E i ( e V ) Be - Mg - FIG. 1. (Color online) Electron affinities of Be − and Mg − ions under the influence of the perturbationpotential (11). Full lines are shown for the exponent α = 0.025, while the broken lines are for α = 0.035.Red color (light gray) describes the Be − ion and the black color is for the Mg − . negative ions used the same basis sets and the same correlation methods (CCSD-T or FCI). Typicaldependence of the electron affinities on external field (11) is shown in Fig. 1 for both negative ions,Be − and Mg − , and in the range of energies used for the present analytic continuation. Fig. 1 yieldsthe following observations: • As expected, the weaker perturbation potential with α = 0.035 requires a stronger scalingparameter λ to achieve the same binding negative ion energies as the perturbation with α =0.025. • Surprisingly, a larger scaling parameter (stronger perturbation) is necessary to bind the Mg − resonance that lies closer to the zero when compared to the Be − resonance (as will be seenbelow). Such behavior may be caused by the spatial extent of the Mg − p resonant wavefunction when compared to the reach of the 2 p wave function of the Be − ion. • The lowest binding energies are not included in the continuation input data because ofthe difficulties we encountered while using the quantum chemistry software. Hartree-Fockmethod is known to destabilize in very diffused basis sets, however low binding energies areinaccurate if a more compact basis is used.Most of the present results were obtained with the CCSD-T method. However, once the theoptimal exponent α (see the Sec. IV) was found for the beryllium atom, the affinity curve shownin Fig. 1 was also recomputed with the expensive FCI method and the basis as described above. IV. RESULTS
As discussed in Sec. II, our goal is to search for regions of stable results with respect to thetwo optimization parameters. First is the range of the input electron affinities defined by maximalaffinity E M . The second parameter, the exponent α in Eq. (11) defines the shape of the perturbationpotential. Typical dependence of the resonance parameters on the maximal energy is shown inFig. 2 for the fixed α parameters. It is clear that the stability is little worse for the Be − ion whencompared to Mg − ion. However, it is possible to narrow the spread of the obtained resonance databy considering the value of χ defined by Eq. (6). Fig. 3 shows the dependence of χ quantity onthe maximal energy E M . A pronounced minimum at E M = 1.92 eV is clearly visible. This allowsan application of a condition of the best fit. Such a restriction leads to a well defined E M foreach choice of the perturbation parameter α producing a data sets shown in Fig 4. For berylliumthe resonance position and width stabilizes for α > α = 0.035 Maximal energy E M used in the fit (eV) R e s on a n ce e n e r gy a nd w i d t h ( e V ) Be - Mg - energywidthenergywidth FIG. 2. (Color online) Resonance energy (shown as circles) and width (displayed as diamonds) calculatedfor Be − and Mg − as functions of the energy extent defined by the maximal energy E M . The exponents α are fixed at α = 0.035 for Be − and α = 0.025 for Mg − . resulting in E r = 0.323 eV and Γ = 0.317 eV. In order to estimate accuracy of the correlation energyprovided by the CCSD-T method we also recomputed this final results with the FCI method. TheFCI affinities yield E r = 0.282 eV and Γ = 0.316 eV. Detailed summary of the available theoreticalresults for the Be − resonance was presented in Tab. III of Ref. [41]. A comparison with the mostrecent computations will be given in Sec. V.In case of magnesium ion the resonance energy is very stable over the whole range of examinedperturbation parameters α . However, the width exhibits a weak dependence on the exponent α .This feature may indicate that the low-order RAC method is inadequate for the Mg − resonance.Nonetheless, the best fit is obtained for α = 0.025, giving E r = 0.188 eV and Γ = 0.167 eV. Theavailable data for the Mg − resonance are summarized in Tab. I. Presently computed resonanceenergy is about 40 meV higher that the experimental value of Burrow et al. [29, 50]. Such adiscrepancy may have several possible reasons:1. The experimental resolution is about 30–40 meV [29].2. Discrepancy between the correlation energies of the CCSD-T and FCI methods and thepresent basis set is about 41 meV for the electron affinity of the beryllium atom. Similardifference can also be expected for the magnesium. Moreover, weaker stability of Γ with re- Maximal energy E M used in the fit (eV) -15 -14 -13 χ ( a . u . ) FIG. 3. (Color online) Quality of the RAC fit for the resonance of Be − as a function of the maximal energy.Exponent α is fixed at 0.035 and the increasing weights set ε = i are used.TABLE I. Comparison of the available data for the resonance energy E r and the resonance width Γ for the3s ε p P state of atomic magnesium.Method Resonance energy E r (eV) Resonance width Γ(eV)Model potential [26] 0.37 0.10Model potential [28] 0.161 0.160Complex rotation [27] 0.08 0.17Stabilization [47] 0.14 0.08Complex SCF [48] 0.50 0.54Finite elements [49] 0.159 0.12Experiment [29] 0.15 ± ∼ spect to the perturbation potential (shown in Fig. 4) indicates that higher order continuationmay be necessary.3. The experimental resonance energy [29] was determined from the maximum of the measuredcross section, whereas present method defines the resonance energy from a pole of the S -0 Exponent α (bohr -2 ) R e s on a n ce e n e r gy a nd w i d t h ( e V ) Be - Mg - FIG. 4. (Color online) Resonance energy E r (circles connected full lines) and the resonance width Γ(diamonds connected with dashed lines) as functions of α parameter of the perturbation potential. matrix. The two definitions give similar results for a narrow resonance (Γ < E r ), but for fora broader resonance (Γ ≥ E r ), as in the present case, the results may differ. V. CONCLUSIONS
Present study confirms the observations of White et al. [19] in which the authors state that theGaussian perturbation potential is more difficult to apply than potentials possessing the Coulombsingularity. It has been shown in the case of a model potential [19] that the trajectory of theresonant pole is more complicated for the Gaussian perturbation. In the present study we haveshown that in order to obtain stable results, the RAC method must be restricted to fairly lowelectron affinities and a careful analysis of the results with respect to the width of the perturbationpotential must be carried out.Such procedure allowed us to apply the RAC method to one of the remaining enigmas amongshape resonances of small atoms, the 2 s εp P resonance of Be − . To the best of our knowledgethere are no experimental data available for this resonance. Important role of the correlation energyin this system creates a challenging task for the theory, albeit the fact that Be − possess only 5electrons. Consequently, about two dozens of theoretical predictions (found in Ref. [31–42]) do1not result in any kind of a consensus. Two methods with high level of correlation description, theCCSD-T and FCI methods, were applied in the present study. While the position of the resonanceshifts to the lower energies by about 41 meV for the more accurate FCI method, the resonancewidth was found insensitive to the correlation treatment. Presently calculated FCI resonant energy E r = 0.282 eV and width Γ = 0.316 eV are in a good agreement with the complex CI results ofMcNutt and McCurdy [35] that predict the E r = 0.323 eV and Γ = 0.296 eV. Recent scatteringcalculations [42] determined the resonance with E r = 0 . ± .
04 eV and Γ = 0 . ± .
06 eV again ina good agreement with the present results. However, another set of recent calculations by Tsedneeet al. [41] place the resonance at E r = 0.756 eV and Γ = 0.874 eV.In case of the 2 s εp P resonance of Mg − a comparison with the experiment is available. Al-though, the present calculations determine the resonance about 40 meV higher than the experiment[29], they still exhibit the best agreement with the experimental data among the ab-initio methods. ACKNOWLEDGMENTS
The contributions of R ˇC were supported by the Grant Agency of the Czech Republic (Grant No.GACR 18-02098S). JH conducted this work with support of the Grant Agency of Czech Republic(Grant No. GACR 16-17230S). IP acknowledges support from the Grant Agency of the CzechRepublic (Grant No. GACR 17-14200S). [1] F. Carelli, M. Satta, T. Grassi, and F. A. Gianturco, Astrophys. J. , 97 (2013).[2] W. F. van Dorp, Phys. Chem. Chem. Phys. , 16753 (2012).[3] R. M. Thorman, R. T. P. Kumar, D. H. Fairbrother, and O. Ing´olfsson, Beilstein J. Nanotechnol. ,1904 (2015).[4] X. Pan, P. Cloutier, D. Hunting, and L. Sanche, Phys. Rev. Lett. , 208102 (2003).[5] Y. Zheng, J. R. Wagner, and L. Sanche, Phys. Rev. Lett. , 208101 (2006).[6] H. S. Taylor and A. U. Hazi, Phys. Rev. A , 2071 (1976).[7] A. U. Hazi and H. S. Taylor, Phys. Rev. A , 1109 (1970).[8] A. U. Hazi, T. N. Rescigno, and M. Kurilla, Phys. Rev. A , 1089 (1981).[9] R. F. Frey and J. Simons, J. Chem. Phys. , 4462 (1986).[10] N. Moiseyev, Phys. Rep. , 212 (1998).[11] C. W. McCurdy and T. N. Rescigno, Phys. Rev. Lett. , 1364 (1978).[12] W. P. Reinhardt, Ann. Rev. Phys. Chem. , 223 (1982). [13] U. V. Riss and H. D. Meyer, J. Phys. B: Atom. Molec. Phys. , 4503 (1993).[14] S. Feuerbacher, T. Sommerfeld, R. Santra, and L. S. Cederbaum, The Journal of Chemical Physics , 6188 (2003).[15] V. I. Kukulin and V. M. Krasnopolsky, J. Phys. A: Math. Gen. , L33 (1977).[16] V. M. Krasnopolsky and V. I. Kukulin, Phys. Lett. A , 251 (1978).[17] V. I. Kukulin, V. M. Krasnopolsky, and J. Hor´aˇcek, Theory of Resonances: Principles and Applications (Kluwer Academic Publishers, Dordrecht/Boston/London, 1988).[18] J. Hor´aˇcek, P. Mach, and J. Urban, Phys. Rev. A , 032713 (2010).[19] A. F. White, M. Head-Gordon, and C. W. McCurdy, J. Chem. Phys. , 044112 (2017).[20] J. Hor´aˇcek, I. Paidarov´a, and R. ˇCur´ık, J. Phys. Chem. A , 6536 (2014).[21] T. Sommerfeld, J. B. Melugin, P. Hamal, and M. Ehara, J. Chem. Theory Comput. , 2550 (2017).[22] P. Papp, ˇS. Matejˇc´ık, P. Mach, J. Urban, I. Paidarov´a, and J. Hor´aˇcek, Chem. Phys. , 8 (2013).[23] R. ˇCur´ık, I. Paidarov´a, and J. Hor´aˇcek, Europ. Phys. J. D , 146 (2016).[24] J. Hor´aˇcek, I. Paidarov´a, and R. ˇCur´ık, J. Chem. Phys. , 184102 (2015).[25] T. Sommerfeld and M. Ehara, J. Chem. Phys. , 034105 (2015).[26] J. Hunt and B. L. Moiseiwitsch, J. Phys. B: Atom. Molec. Phys. , 892 (1970).[27] P. Krylstedt, M. Rittby, N. Elander, and E. Brandas, J. Phys. B: Atom. Molec. Phys. , 1295 (1987).[28] L. Kim and C. H. Greene, J. Phys. B: Atom. Molec. Phys. , L175 (1989).[29] P. D. Burrow, J. A. Michejda, and J. Comer, J. Phys. B: Atom. Molec. Phys. , 3225 (1976).[30] S. J. Buckman and C. W. Clark, Rev. Mod. Phys. , 539 (1994).[31] H. A. Kurtz and Y. ¨Ohrn, Phys. Rev. A , 43 (1979).[32] H. A. Kurtz and K. D. Jordan, J. Phys. B: Atom. Molec. Phys. , 4361 (1981).[33] C. W. McCurdy, T. N. Rescigno, E. R. Davidson, and J. G. Lauderdale, J. Chem. Phys. , 3268(1980).[34] T. N. Rescigno, C. W. McCurdy, and A. E. Orel, Phys. Rev. A , 1931 (1978).[35] J. F. McNutt and C. W. McCurdy, Phys. Rev. A , 132 (1983).[36] P. Krylstedt, N. Elander, and E. Brandas, J. Phys. B: Atom. Molec. Phys. , 3969 (1988).[37] Y. Zhou and M. Ernzerhof, J. Phys. Chem. Let. , 1916 (2012).[38] A. Venkatnathan, M. K. Mishra, and H. J. A. Jensen, Theor. Chem. Acc. , 445 (2000).[39] K. Samanta and D. L. Yeager, J. Phys. Chem. B , 16214 (2008).[40] K. Samanta and D. L. Yeager, Int. J. Quantum Chem. , 798 (2010).[41] T. Tsednee, L. Liang, and D. L. Yeager, Phys. Rev. A , 022506 (2015).[42] O. Zatsarinny, K. Bartschat, D. V. Fursa, and I. Bray, J. Phys. B: Atom. Molec. Phys. , 235701(2016).[43] P. J. Knowles, C. Hampel, and H. Werner, J. Chem. Phys. , 5219 (1993).[44] M. J. Deegan and P. J. Knowles, Chem. Phys. Lett. , 321 (1994).[45] H. J. Werner, P. J. Knowles, R. Lindh, F. R. Knizia, F. R. Manby, M. Sch¨utz, and Others, MOLPRO, version 2010.1, a package of ab initio programs (2010).[46] B. P. Prascher, D. E. Woon, K. A. Peterson, T. H. Dunning, and A. K. Wilson, Theor. Chem. Acc. , 69 (2011).[47] J. S. Chao, M. F. Falcetta, and K. D. Jordan, J. Chem. Phys. , 1125 (1990).[48] C. W. McCurdy, J. G. Lauderdale, and R. C. Mowrey, J. Chem. Phys. , 1835 (1981).[49] G. A. Gallup, Phys. Rev. A , 012701 (2011).[50] P. D. Burrow and J. Comer, J. Phys. B: Atom. Molec. Phys.8