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Multi-Reference Quantum Chemistry Protocol for Simulating Autoionization Spectra: Test of Ionization Continuum Models for the Neon Atom

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Multi-Reference Quantum Chemistry Protocol for Simulating Autoionization Spectra: Test of Ionization Continuum Models for the Neon Atom Multi-reference quantum chemistry protocol for simulating autoionization spectra: Test of ionization continuum models for the neon atom Gilbert Grell, Oliver K¨uhn,and Sergey I. Bokarev∗ Institut f¨urPhysik, Universit¨atRostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany (Dated: August 28, 2019) In this contribution we present a protocol to evaluate partial and total Auger decay rates combin- ing the restricted active space self-consistent field electronic structure method for the bound part of the spectrum and numerically obtained continuum orbitals in the single-channel scattering theory framework. On top of that, the two-step picture is employed to evaluate the partial rates. The performance of the method is exemplified for the prototypical Auger decay of the neon 1s−13p reso- nance. Different approximations to obtain the continuum orbitals, the partial rate matrix elements, and the electronic structure of the bound part are tested against theoretical and experimental ref- erence data. It is demonstrated that the partial and total rates are most sensitive to the accuracy of the continuum orbitals. For instance, it is necessary to account for the direct Coulomb potential of the ion for the determination of the continuum wave functions. The Auger energies can be repro- duced quite well already with a rather small active space. Finally, perspectives of the application of the proposed protocol to molecular systems are discussed. I. INTRODUCTION highly charged cations while a cascade of highly reactive low energy electrons is emitted [11, 14{17]. Further, in Ionization triggered by photon absorption occurs along free electron laser experiments operating with ultrashort two pathways. In direct photoionization, the energy is intense X-ray pulses, autoionization after multiple pho- transferred to an ejected electron. Alternatively, the sys- toionization induces the Coulomb explosion of the target, tem can be first put into a metastable state by a reso- which limits the achievable spectroscopic and temporal nant excitation and afterwards decay via an autoioniza- resolution [18]. Due to this wealth of applications, au- tion mechanism. Autoionization can be approximately toionization and especially the local Auger effect have understood as a two-step process [1], in which the de- been studied extensively both theoretically and experi- cay can be considered independently from the excitation mentally since its discovery by Meitner [5] and descrip- process and interferences between direct and autoion- tion by Wentzel [1]. ization are neglected. For example, let us consider an atomic species, such as a neon atom that is prepared in a highly excited state Ψi above the continuum threshold 860 at E = 0 eV, Fig. 1. Thisj i state spontaneously decays into Ψi Ψα i 850 the continuum state Ψα comprising the discrete state | i| i E core + j i core Ψ of the ion and the emitted electron, , carry- ψα εα f j αi | i ing theE excess energy "α = i f . The system's elec- E − E 40 tronic structure is thus encoded into the kinetic energy spectrum of the ionized electrons. Photoelectron Spec- 30 troscopy (PES) and Autoionization Spectroscopy (AIS) + Ψf f 20 map bound states to the continuum which makes them | i E continuum E (eV) valence less sensitive to selection rule suppression and more infor- valence 10 mative than spectroscopies involving optical transitions between bound states [2{4]. 0 Autoionization processes, predominantly Auger de- 10 cay [5], but also Interatomic Coulombic Decay (ICD) [6] − and Electron Transfer Mediated Decay (ETMD) [7] are 20 arXiv:1905.05785v2 [physics.chem-ph] 27 Aug 2019 − particularly interesting on their own. Due to their cor- Ne Ne+ related nature, they not only probe but also initiate or compete with intricate ultrafast electronic and nuclear Figure 1. Autoionization scheme for the neon atom. The core dynamics e.g. [8{12]. Additionally, they provide the vacancy state jΨii with energy Ei (red) decays isoenergetically main channel for the decay of core vacancies [13] and into the continuum state jΨαi (black) composed of the ionic E play a key role in biological radiation damage, creating + bound state Ψf with energy Ef (blue) and the continuum orbital j αi of the outgoing electron with the excess energy "α. States that do not contribute to the process are depicted in gray; the singly ionized continuum is denoted by the color ∗ [email protected] gradient. 2 Remarkably, AIS simulations of molecular systems re- most general and accurate quantum-mechanical treat- main challenging until today, although the fundamen- ment of the problem, thus potentially serving as a high- tal theory is known for decades [19{21]. For atoms, level reference, although connected to substantial com- methods combining highly accurate four-component putational effort. Multi-configurational Dirac-Fock (MCDF) calculations with multichannel scattering theory are publicly avail- Summarizing, most of the mentioned methods have able [22], whereas no such general purpose code exists been applied only to simple diatomics, first row hydrides, for molecules. The main complication of the molecu- halogen hydrides, and small molecules consisting of not lar case lies in the construction of molecular continuum more than two heavy atoms. Studies of larger molecular states Ψα . The approaches to the simulation of AIS systems, such as tetrahedral molecules, small aldehydes, publishedj duringi the last decades can be classified into and amides [56, 57], solvated metal ions [11] and poly- two families { those that circumvent the continuum or- mers [58] are very scarce. In fact, the Fano-ADC [32, 33] bital problem and those that treat the continuum orbital and the XMOLECULE [40, 41] approaches are the only explicitly. publicly available tools that allow to simulate AIS for a The first family comprises the following flavors: The variety of systems without restricting the molecular ge- simplest method that allows to assign experimental AIS ometry. Further, both methods are not suited to treat is to evaluate the energetic peak positions [23{25]. On systems possessing multi-configurational wave functions. top of that simple estimates for the partial decay rates This puts studies of some chemically interesting systems can be obtained based on an electron population analy- having near-degeneracies, for example, transition metal sis [26]. More advanced approaches rely on an implicit compounds, or of photodynamics in the excited electronic continuum representation with Stieltjes imaging [27], a states, e.g., near conical intersections, out of reach. To Green's operator [28, 29], or a propagator [30, 31] formal- keep up with the experimental advancements, the de- ism. From this group, the Fano-Stieltjes Algebraic Di- velopment of a general purpose framework to evaluate agrammatic Construction (Fano-ADC) method [32] has autoionization decay rates (Auger, ICD, and ETMD) been used to evaluate Auger, ICD and ETMD decay rates for molecular systems is warrant. Such a framework of van der Waals clusters [33], first row hydrides [34] and should be kept accessible, transferable and easy to use, 2+ i.e. it should be based on widespread robust and versatile the [Mg(H2O)6] cluster [11]. Therein, the continuum is approximately represented with spatially confined basis Quantum Chemistry (QC) methods. functions. However, the description of Auger electrons Here, we present a protocol that combines multi- with kinetic energies of several hundreds of eV requires configurational Restricted Active Space (RAS) Self- large basis sets leading to computationally demanding Consistent Field (SCF) (RASSCF) bound state wave simulations. functions with single-centered numerical continuum or- The second family, relying on an explicit representa- bitals in the single-channel scattering theory framework tion of the continuum wave function, consists of the fol- [59]. We have chosen the RASSCF approach, since it is lowing approaches: The one-center approximation which known to yield reliable results for core-excited states [60], uses atomic continuum functions centered at the vacancy- needed in the simulation of X-ray absorption [61, 62], bearing atom to describe the outgoing electron in the resonant inelastic scattering [63, 64],and photoemission evaluation of partial decay rates [35{38]. This ap- spectra [65{67] suggesting its application to AIS. proximation can be applied on top of high-level elec- tronic structure methods [39]. It is also applied in the Although the ultimate goal is to investigate molecules, XMOLECULE package [40, 41] which is based on very this proof-of-concept contribution focuses on the simu- cost efficient electronic structure calculations and atomic lation of the prototypical neon 1s−13p Auger Electron continuum functions. This allows for the evaluation Spectrum (AES) to calibrate the approach, since highly of ionization cascades but may limit the applicability accurate reference data are available from both theory for strongly correlated systems, e.g., possessing a multi- [68] and experiment [69{72]. Special attention is paid to configurational character. Further, the influence of the representation of the radial continuum waves, which the molecular field may be taken into account pertur- is investigated herein by a thorough test of different ap- batively [42, 43] or in a complete manner with, e.g., the proximations. Note that our implementation allows to single-center approach, where the whole molecular prob- calculate molecular AIS as well as PES which will be lem is projected
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