Symmetry Breaking and Hole Localization in Multiple Core Electron

Article pubs.acs.org/JPCA

Symmetry Breaking and Hole Localization in Multiple Core Electron Ionization V. Carravetta*,† and H. Ågren‡ †

CNR-IPCF, Institute of Chemical and Physical Processes, via G. Moruzzi 1, I-56124 Pisa, Italy, and Department of Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, SE-106 91 Stockholm, Sweden

‡

ABSTRACT: Motivated by recent opportunitites to study hollow molecules with multiple core holes offered by X-ray free electron lasers, we revisit the core-hole localization and symmetry breaking problem, now studying ionization of more than one core electron. It is shown, using a N2 molecule with one, two, three, and four core holes, for example, that in a multiconfigurational determination of the core ionization potentials employing a molecular point group with broken inversion symmetry, one particular configuration is sufficient to account for the symmetry breaking relaxation energy in an independent particle approximation in the case of one or three holes, whereas the choice of point group symmetry is unessential for two and four holes. The relaxation energy follows a quadratic dependence on the number of holes in both representations.

■

INTRODUCTION The creation of double- or multiple-core vacancies has been a rather exotic phenomenon that until recently has attracted only little attention. However, such species were early observed indirectly in cascade and inner shell Auger or Coster−Kronig processes,1 where the spectral features are faint and broadened by the short lifetimes of the deeper core levels involved. Double-core-hole states have also appeared in X-ray hypersatellite emission 2 and one-photon two-electron X-ray absorption3 spectra of noble gases. Except for the noble gases cases the observation of these processes has referred mostly to metallic elements.4 Through high-resolution synchrotron radiation techniques, the conditions for their generation in absorption spectroscopy were obtained already some 20 years ago.5 With the brilliance of fourth generation synchrotron sources, X-ray free electron lasers, the conditions for their detailed study will be substantially increased.6−8 The problem of localization and symmetry breaking, dating back to Löwdin’s so-called symmetry dilemma for Hartree− Fock,9 has been a topic of general interest for spectroscopy. As the independent particle model (Hartree−Fock) is often dictating our perception of electronic structure, this symmetry dilemma has become particularly conspicuous for the interpretation of X-ray spectra involving atom centered core electrons and was accordingly studied in a number of early papers on X-ray and X-ray photoelectron spectra.10−16 A landmark paper was that of Bagus and Schaeffer17 who could show that a Hartree−Fock calculation of the core ionization potentials of O2 was about 15 eV in error with the experiment when full molecular symmetry D∞h was imposed (delocalized core orbitals), whereas the error decreased to only 1 eV in the case of C∞v symmetry (localized core orbitals). In another key paper, by Cederbaum and Domcke, the symmetry breaking was analyzed by decomposition of the ionization potential into © 2013 American Chemical Society

relaxation and correlation contributions using a Green's function approach.11 Evidence on the merits of a localized picture was given by subsequent studies of other core-hole state related properties, such as energy gradients, vibronic coupling constants,15 electronic transition moments,18 angular distributions,19 and satellites.20 The symmetry dilemma was proven remedied if one goes beyond the independent particle picture and employs, in the high symmetry, a limited configuration interaction calculation, which compensates the missing electron relaxation, O2+ being a prime example.16 Although the symmetry breaking in a homonuclear diatomic molecule does not refer to the exact wave function, but merely to an approximation of it, symmetry breaking and core-hole localization physically occurs in polyatomic molecules containing an element of symmetry, as an effect of vibronic coupling between quasi-degenerate symmetry adapted core levels. Such “pseudo Jahn−Teller” effect leads to a lowering of the point group, the classical example, given by Domcke and Cederbaum,11,12 is the O1 s−1 Σu and Σg states of CO2+ that couple over the antisymmetric stretching modes, thereby lowering the symmetry from D∞h to C∞v.12 Thus even if the electronic Hamiltonian cannot break the symmetry, a vibronic Hamiltonian in a diabatic electronic representation easily displays such effects. With the availability of XFELs and coming multiple core-hole studies, coupled to new insight of those states offered by computational analysis,21−23 we find it of interest to study how Löwdins symmetry dilemma appears for such states and how the pattern of core-hole state relaxation and configurational interaction may arise; the N2 molecule being here an obvious prime test case. Received: July 4, 2013 Published: July 16, 2013 6798

dx.doi.org/10.1021/jp406602y | J. Phys. Chem. A 2013, 117, 6798−6802

The Journal of Physical Chemistry A

Article

Table 1. Binding Energy (eV) for Single- (h), Double- (hh), Triple- (hhh), and Quadruple-Core-Hole (hhhh) States of N2 at Different Computational Levels: SCF Frozen Orbitals (f), SCF Relaxed Orbitals (r), Restricted Active Space MCSCF (RAS)a holes

BEf

BEr

Er

Ec

BErc

BEπu−πg

BERAS

exp

hg hu hloc hhg hhu hhSloc hhTloc hhhg hhhu hhhloc hhhh

425.8 425.7 425.7 865.8 865.6 865.7 865.7 1417.9 1418.0 1417.9 1984.4

418.6 418.3 409.9 837.3 837.1 837.1 837.1 1358.5 1358.4 1348.7 1883.2

−7.3 −7.3 −15.8 −28.5 −28.6 −28.6 −28.6 −59.4 −59.6 −69.2 −101.2

−0.8 −0.8 −0.8 −2.7 −2.7 −2.7 −2.7 −3.5 −3.5 −3.5 −7.3

417.7 417.5 409.1 834.5 834.4 834.4 834.4 1355.0 1355.0 1345.3 1875.9

410.2 410.0

409.9 409.7 408.9 834.1 834.7 834.1 834.7 1346.3 1346.3 1345.7 1878.4

409.9

1347.5 1347.5

a Gerade (g) and ungerade (u) subscripts refer to D2h symmetry, whereas loc subscript refers to C2v symmetry. S and T superscripts indicate singlet and triplet coupling of the two holes, respectively. Relaxation energy is denoted by Er; correlation energy contribution to the binding energy by MP2 approximation is denoted by Ec.

■

CALCULATIONS For multiple-, as for single-, core-hole states, there are some particularities, associated with their calculation, that set restrictions on the choice of computational methodology with respect to normal multiple-valence-hole states. Core-hole states are subject to variational collapse, and in a configuration interaction (CI) calculation their root index, although finite, is unpredictable a priori. For wave functions, like MCSCF, that are parametrized in terms of orbitals (orbital rotations) and electronic configurations (determinants) a way to handle this problem is to apply a sequential-step orbital optimization procedure while restricting electron occupancy (restricted active space, RAS) with the desired and fixed core orbital occupancy in the configurations space. It is in most cases sufficient to first freeze core orbitals and then allow all orbital relaxation, to create and maintain at full relaxation the correct local energy minimum for the core ionized state; see ref 24 and early applications for single-core-hole states in, e.g., ref 25, and more recent applications on double-core valence hole states in refs 26 and 27. In the configurational space the variational collapse is avoided by imposing occupancy restrictions for the core orbitals as the interaction between states with empty, single, or doubly occupied core orbitals is negligible by virtue of their huge energy separation. This can be accomplished through using the restricted active space (RAS) technology,28 with a separated subspace allocated to core orbitals. Adopting the lower symmetry, the localization of the core orbitals is forced in a precalculation, followed by the freeze−relax RAS procedure (see above) that maintains localization. All calculations are executed with DALTON program.29

effect and partially the correlation effect. Finally, the seventh column contains an estimation of the BE obtained by RAS calculations, including up to quadruple excitations from valence and inner valence orbitals. In Table 2 we present a basis set Table 2. Binding Energy (eV) for Single-, Double-, Triple-, and Quadruple-Core-Hole States of N2 in Frozen and Relaxed Approximationa with Different Basis Sets core hole hg hu hhg hhu hhhg hhhu hhhh hg hu hhg hhu hhhg hhhu hhhh a

cc-pVDZ

cc-pVTZ

cc-pVQZ

Frozen Approximation 426.10 425.84 425.84 425.92 425.66 425.65 866.30 865.78 865.78 866.17 865.65 865.65 1418.73 1417.92 1417.92 1418.79 1417.99 1417.99 1985.48 1984.39 1984.40 Relaxed Approximation 419.89 418.57 418.43 419.69 418.34 418.21 843.02 837.27 836.67 842.86 837.09 836.48 1370.53 1358.51 1357.69 1370.56 1358.43 1357.78 1906.84 1883.22 1881.95

cc-pV5Z 425.85 425.66 865.80 865.67 1417.95 1418.02 1984.43 418.38 418.16 836.45 836.26 1357.31 1357.40 1881.22

See also caption of Table 1.

study of BE for states with different numbers of core holes in the localized and delocalized representations. For this study four sets of basis functions, cc-pVXZ with X = D, T, Q, 5 of comparable character but of increasing size, from D (double) to 5 (quintuple) quality, were employed. They are distinguished by the letters “cc” (correlation-consistent) and are considered particularly suitable for MCSCF calculations. They are available in the database of the program DALTON, where all the specific references can also be found. The extension of the basis for each N atom varies from [3s, 2p, 1d] (DZ) to [6s, 5p, 4d, 3f, 2g, 1h] (5Z) contracted Gaussian functions. The dependence of the results of quantum calculations by the quality of the basis set is a matter of concern when the target is very accurate calculations. This dependence can, of course, be different depending on the moleculare property that is evaluated and using variational methods generally tends to decrease with the

■

RESULTS AND DISCUSSION In Table 1 we have collected all computed binding energies (BE) for the one-, two-, three-, and four-core-hole states of N2 with D2h symmetry delocalized and C2v symmetry localized representations of the core orbitals. We show values obtained in the frozen orbital approximation and from Hartree−Fock optimization including electron relaxation in the second and third columns, respectively. The energy difference, relaxation energy, is reported in the fourth column, whereas an estimation of the correlation energy contribution to the BE, computed by the MP2 approximation, is reported in column 5. The next column presents the BE values, including fully the relaxation 6799

dx.doi.org/10.1021/jp406602y | J. Phys. Chem. A 2013, 117, 6798−6802

The Journal of Physical Chemistry A

Article

corrected by the hole−hole interaction parameter. For doublecore spectra pertaining to different sites the exchange interaction Kij is often small, and the hole−hole interaction energy is then simply given by the classical hole−hole repulsion depending on the inverse interatomic distance. An analysis of the repulsion and relaxation energies in the case of two-hole molecular states was earlier performed by a perturbation theory approach35 and much more recently reconsidered by DFT,22 MRCI,23 and CASSCF21 calculations. These last investigations mainly focused on the evaluation of the dynamic, also termed interatomic, relaxation, which is the difference between the total relaxation and the sum of the one-hole relaxation energies. As earlier shown in ref 26, it can vary quite a lot for a polyatomic molecule depending on the particular geometric and electronic structure. For the two-hole one-site case the dynamic relaxation is always strongly negative (quadratic with respect to number of holes applying Snyder’s analysis34), whereas for the two-hole two-site case it can also be positive, as indicated in the case of formamide in ref 26. For N2, we get the apparent result that in the symmetry adapted case the dynamic relaxation has a large negative value (about −14 eV), whereas it is predicted to be positive (+3 eV) when symmetry breaking is allowed. This, however, is due only to the fact that the two-hole final state is virtually the same (one hole on either site) in the symmetry adapted and symmetry broken cases, whereas the one-hole relaxation values experience strong symmetry breaking. Nevertheless, the relaxation toward atomic sites seems to interfere negatively in the case of the nitrogen molecule. We also note that the energy splitting between singlet (S) and triplet (T) states, and thus exchange interaction energy, is completely negligible for two-core-hole ionization of N2. As seen in Table 1, the hhg, hhu, and hhloc (S or T) ionization potentials are virtually the same in both frozen and relaxed pictures. In fact, the lowest state in either scheme allocates the two holes at two sites. The placement of the holes on different sites is easily understood in the delocalized orbital picture, because a double hole on the same atom is clearly energetically unfavorable. Let us instead consider the relaxed HF wave functions with the full symmetry. In the case of triplet coupling for the two core electrons, there is just one configuration with one electron in 1σg and one electron in 1σu. In the case of singlet coupling there are instead two configurations; one with two electrons in 1σg and two holes in 1σu and the other one with the reversed occupancy. In this case the configuration with two holes in the same orbital has a low energy because the core orbital is delocalized. Then also the two holes are delocalized and the relaxation energy of the remaining electrons in the field of such a delocalized double hole is very close to that in the field of one hole in each localized core orbital. Three-Hole States. In the three-hole case we see, for the relaxed values, an interesting analogy with the one-hole case: a missing relaxation of roughly 10 eV when a delocalized orbital picture is adopted. In a further analogy with the one-hole case, −2 when the triple-core-hole configuration 1σ−1 g 1σu is interacting with the single excited near-degenerate 1σg−21σu−11πu−1 1πg1 configuration, one recovers almost all of the missing relaxation. Thus the notion that a relaxation effect in the localized picture is translated to a configuration interaction effect in the delocalized picture also holds here. However, like the onehole case this is an MCSCF effect (including orbital relaxation) rather than a CI effect, because the latter one retains only 3 eV of the missing relaxation. Thus, although the total relaxation is

size of the basis set. In the case of core binding energy of N2, the data in Table 2 show that a satisfactory convergence of the results is reached rather quickly. One-Hole States. The one-hole ionization case (h) is listed in the top panel of Table 1. One first notices a rewarding agreement of the localized relaxed calculation with the experimentally determined single-core ionization potential 409.9 eV,30 the only one presently available for N2. Such an agreement should evidently be qualified by neglect of basis set convergence (an energy of about 0.1 eV, as noted in Table 2), by missing correlation energy (indeed correlated values, RAS and MP2, reported in columns 6 and 5, respectively, give somewhat worse agreement), by neglect of relativistic effects (about +0.2 eV31), and by zero-point vibration correction (about −0.2 eV32). However, these small energy contributions are not of interest and are not essential for our further analysis. There is, moreover, no particular physical argument that the one-hole agreement obtained in the relaxed localized calculation does not hold also for the multihole cases given below, although we indeed notice an increased basis set dependence of the relaxed calculations with the number of holes. One may observe the characteristically small splitting between the symmetry adapted core orbital (1σg and 1σu) energies in column 2 of Table 1. Following the analyis of Kosugi33 this splitting mainly originates in (symmetry dependent) differential core valence penetration for the two core orbitals. Comparing frozen and relaxed results in Table 1, we find the characteristics of a roughly double relaxation energy (−16 eV) in the localized scheme with respect to the delocalized scheme (−7 eV). This accords with the analysis of Snyder et al. using Slater’s screening rules34 on a quadratic dependence of the relaxation energy on degree of shielding. When the single-core-hole configuration 1σ−1 g is allowed to −1 1 interact with the single excited near-degenerate 1σ−1 u 1πu 1πg configuration in the MCSCF calculation, one obtains a binding energy of 410.2 eV, thus retaining practically all of the missing relaxation. This result conforms with the notion that a relaxation effect in the localized picture is translated to a configuration interaction effect in the delocalized one. The numerical results and physical picture are thus compatible with the early work for O2 by Ågren, Bagus, and Roos.16 We note, however, that if the interaction is considered at the CI level only, i.e., projecting on the 1σ−1 g state orbitals, only a small fraction, about 3 eV, of the missing relaxation energy is obtained, and thus that the MCSCF procedure with a coupled orbital optimization is essential for a full recovery. Adding more dynamic electron correlation through the RAS or MP2 approaches, one certainly modifies the final comparison, but without changing the qualitative picture. Two-Hole States. In the case of two holes i, j the total core BE can be partitioned as Eij = εi + εj + R ijS,T + V ijS,T

(1)

where εi and εj denote single ionization energies with holes in S,T the frozen core orbitals i and j. RS,T ij is a relaxation term and Vij a hole−hole interaction term, both dependent on the spincoupling of the two-hole state. In terms of the Coulomb J and exchange K integrals, VS,T = Jij ± Kij for singlet and triplet ij coupling and VSii = Jii. In an approximation corresponding to Koopmans’ level of approximation for photoelectron spectra, Rij is assumed zero. The double-hole ionization potentials are then simply related to sum of two (negative) orbital energies 6800

dx.doi.org/10.1021/jp406602y | J. Phys. Chem. A 2013, 117, 6798−6802

The Journal of Physical Chemistry A

Article

Notes

almost a factor of 4 larger in the three-hole than in the one-hole case, the same numerics and the same physical picture of the symmetry breaking and restoration is obtained in the two cases. We also see that here, as for the other cases with different numbers of core holes, the addition of dynamical valence correlation is not essential (see column 5 of Table 1), as this contribution tends to compensate between the ground and final states. The negative (but small) MP2 contribution indicates that valence correlation energy is slightly larger in the more compressed electron cloud of the core ionized states than in the ground state (core−core correlation in the ground state is not accounted for). Four-Hole States. A four-core-hole nitrogen molecule is a highly energetic hollow species, with presumably very low probability of ever being observed, but with some very interesting exotic chemical and physical properties unmatched by anything available in the laboratory. Here we notice a 100 eV relaxation energy upon core ionization, thus 10 times as large than in the one-hole case, almost 4 times that of the double-hole case, and almost twice the relaxation of the triplehole case. Snyder’s analysis of a quadratically dependent relaxation energy on the degree of shielding of the potential seems to be verified. The symmetry dilemma is evidently of no consequence in this case because there are no core electrons left to localize.

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Jeppe Olsen and Olav Vahtras for assistance with the code. This research was partly supported by PRIN 2010ERFKXL.

■

■

CONCLUSIONS As demonstrated in this work, the unconstrained MCSCF technique24,26 can be used to obtain symmetry restricted and symmetry broken solutions to the multiply core ionization energies as well as multiple core excitation energies. We capitalized on this in the present work and studied the N2 molecule to test the notion of symmetry breaking for multiple core-hole states. We find that the incompleteness of the oneparticle picture and Löwdin’s symmetry dilemma easily can be solved with a restricted MCSCF approach where the symmetry breaking relaxation in the lower symmetry point group is remedied by a particular “doubly excited” configuration in the higher symmetry point group. This configuration describes a core−core excitation with symmetry reflection coupled to a valence−valence excitation with symmetry reflection. However, CI catches only a fraction of the missing relaxation energy in the high point group symmetry when compared to MCSCF including orbital optimization. A nice similarity, numerically and physically, of this interaction is found between the cases of single- and triple-hole ionization, although the total relaxation effect is much larger in the latter case. For double holes the lowest states are identical in symmetry restriction and symmetry unrestriction already in the independent-particle representation. N2 with completely empty shells can also be readily optimized by the applied unrestricted MCSCF scheme. This hollow species shows superlarge relaxation with 10 times a normal one-core-hole relaxation energy, however, obviously without any bearing on the symmetry dilemma. As density functional theory suffers from the same type of symmetry dilemma as Hartree−Fock, we predict that newly developed integrated MCSCF-DFT (MCr-DFT) techniques are the most promising for exploring with high precision multiple core-hole states of larger species containing an element of symmetry.

■

REFERENCES

(1) Siegbahn, H. Excited States in Quantum Chemistry; D. Reidel Publishing Co.: Dordrecht, The Netherlands, 1978. (2) Ågren, H.; Nordgren, J.; Selander, L.; Nordling, C.; Siegbahn, K. Multiplet Structure in the High-Resolution X-Ray Emission Spectrum of Neon. J. Electron Spectrosc. Relat. Phenom. 1978, 14, 27. (3) Deslattes, R.; La VIlla, R.; Cowan, P.; Henis, A. Threshold Studies of a Multivacancy Process in the KSS Region of Argon. Phys. Rev. A 1983, 27, 923. (4) Deutsch, M.; Hart, M. Single X-Ray Photon Excitaton of Two and Three Electrons in Nickel. Phys. Rev. A 1984, 29, 2946. (5) Bodeur, S.; Reynaud, C.; Bisson, K.; Millie, P.; Nenner, I.; Rockland, U.; Baumgärtel, H. In Proceedings of the International Conference on Synchrotron Radiation and Dynamic Phenomena, Grenoble, France; Beswick, A., Ed.; American Intsitute of Physics: Melville, NY, 1991. (6) Eland, J. H. D.; L., P.; E., M.; U., K.; Tashiro, M.; Feifel, R. Double Core Hole Creation and Subsequent Auger Decay in NH3 and CH4. Phys. Rev. Lett. 2010, 105, 213005. (7) Ishikawa, T.; et al. A Compact X-Ray Free-Electron Laser Emitting in the Sub-Ångström Region. Nat. Photonics 2012, 6, 540. (8) Vartanyants, I. A.; et al. First Lasing and Operation of an Ångstrom-Wavelength Free-Electron Laser. Phys. Rev. Lett. 2011, 107, 144801. (9) Löwdin, P. Rev. Mod. Phys. 1963, 35, 469. (10) Jonkman, H.; van der Welde, G.; Nieuwport, W. Ab Initio SCF MO Calculation of Ionisation Energies and Charge Distributions of TCNQ and Its Mono- and Divalent Anions. Chem. Phys. Lett. 1974, 25, 62. (11) Cederbaum, L. S.; Domcke, W. Localized and Delocalized Core Holes and Their Interrelation. J. Chem. Phys. 1977, 66, 5084. (12) Domcke, W.; Cederbaum, L. S. Vibronic Coupling and Symmetry Breaking in Core Electron Ionization. Chem. Phys. 1976, 25, 189. (13) Butscher, W.; Buenker, R.; Peyerimhoff, S. All-Electron CI Calculations for Core-Ionized, Core-Valence Excited and Shake-up States of N2. Chem. Phys. Lett. 1977, 52, 449. (14) Canuto, S.; Goscinski, O.; Zerner, M. Broken Orbital Symmetry Study of Low-Lying Excited and N15 Ionized States of Pyrazine. Chem. Phys. Lett. 1979, 68, 232. (15) Müller, J.; Ågren, H.; Goscinski, O. Role of Localization in the Prediction of Core ESCA Lineshapes of N2, O2 and NO. Chem. Phys. 1979, 38, 349. (16) Ågren, H.; Bagus, P.; Roos, B. Symmetry Adapted versus Symmetry Broken Wavefunctions: the 1s Core Level Ions of O+2. Chem. Phys. Lett. 1981, 82, 505. (17) Bagus, P.; Schaeffer, F. Chem. Phys. Lett. 1981, 82, 505. (18) Ågren, H.; Nordgren, J. Theor. Chim. Acta 1981, 58, 111. (19) Dill, D.; Wallace, S.; Siegel, J.; Dehmer, J. MolecularPhotoelectron Angular Distributions as a Probe of Dynamic Symmetry Breaking. Phys. Rev. Lett. 1978, 41, 1230. (20) Arneberg, R.; Müller, J.; Manne, R. Configuration Interaction Calculations of Satellite Structure in Photoelectron Spectra of H2O. Chem. Phys. 1982, 64, 249. (21) Tashiro, M.; Ehara, M.; Fukuzawa, H.; Ueda, K.; Buth, C.; Kryzhevoi, N. V.; Cederbaum, L. S. Molecular Double Core Hole Electron Spectroscopy for Chemical Analysis. J. Chem. Phys. 2010, 132, 184302.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 6801

dx.doi.org/10.1021/jp406602y | J. Phys. Chem. A 2013, 117, 6798−6802

The Journal of Physical Chemistry A

Article

(22) Ueda, K.; Takahashi, O. Extracting Chemical Information of Free Molecules from K-Shell Double Core-Hole Spectroscopy. J. Electron Spectrosc. Relat. Phenom. 2012, 185, 301−311. (23) Kryzhevoi, N. V.; Tashiro, M.; Ehara, M.; Cederbaum, L. S. Interatomic Relaxation Effects in Double Core Ionization of Chain Molecules. J. Chem. Phys. 2012, 137, 154316. (24) Jensen, H.; Jørgensen, P.; Ågren, H. Efficient Optimization of Large Scale MCSCF Wave Functions with a Restricted Step Algorithm. J. Chem. Phys. 1987, 87, 451. (25) Naves de Brito, A.; Svensson, S.; Correia, N.; Ågren, H. A Theoretical Study of X-Ray Photoelectron Spectra of Model Molecules for Polymethylmethacrylate. J. Chem. Phys. 1991, 95, 2965. (26) Ågren, H.; Jensen, H. Relaxation and Correlation Contributions to Molecular Double Core Ionization Energies. Chem. Phys. 1993, 172, 45. (27) Andersson, E.; Niskanen, J.; Hedin, L.; Eland, J.; Linusson, P.; Karlsson, L.; Rubensson, J.-E.; Carravetta, V.; Ågren, H.; Feifel, R. Core-Valence Double Photoionization of the CS2 Molecule. J. Chem. Phys. 2010, 133, 094305. (28) Jensen, H.; Ågren, H.; Olsen, J. In Modern Techniques in Computational Chemistry: MOTECC-90; Clementi, E., Ed.; Escom: Leiden, 1990; p 435. (29) DALTON, a molecular electronic structure program, Release 2.0 (2005); see http://daltonprogram.org/. (30) Siegbahn, K.; Nordling, C.; Johansson, G.; Hedman, J.; Hedén, P. F.; Hamrin, K.; Gelius, U.; Bergmark, T.; Werme, L. O.; Manne, R.; Baer, Y. ESCA Applied to Free Molecules; North-Holland Publishing Co.: Amseterdam, 1969. (31) Pekeris, C. Ground State of Two-Electron Atoms. Phys. Rev. 1958, 112, 1649. (32) Müller, J.; Ågren, H. In Molecular Ions; Berkowitz, J., Groenewald, K.-O., Eds.; Plenum Publishing Corp.: New York, 1983. (33) Kosugi, N. Spin-Orbit and Exchange Interactions in Molecular Inner Shell Spectroscopy. J. Electron Spectrosc. Relat. Phenom. 2004, 137, 335. (34) Snyder, L. Core-Electron Binding Energies and Slater Atomic Shielding Constants. J. Chem. Phys. 1971, 55, 95. (35) Cederbaum, L. S.; Tarantelli, F.; Sgamellotti, A.; Schirmer, J. On Double Vacancies in the Core. J. Chem. Phys. 1986, 85, 6513.

6802

dx.doi.org/10.1021/jp406602y | J. Phys. Chem. A 2013, 117, 6798−6802