Molecular double core-hole electron spectroscopy for chemical analysis

Molecular double core-hole electron spectroscopy for chemical analysis

Motomichi Tashiro    Masahiro Ehara ehara@ims.ac.jp Institute for Molecular Science, Okazaki 444-8585, Japan    Hironobu Fukuzawa    Kiyoshi Ueda Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan    Christian Buth The PULSE Institute for Ultrafast Energy Science, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA    Nikolai V. Kryzhevoi    Lorenz S. Cederbaum Theoretical Chemistry, Institute of Physical Chemistry, Heidelberg University, 69120 Heidelberg, Germany
July 13, 2019
Abstract

We explore the potential of double core hole electron spectroscopy for chemical analysis in terms of x-ray two-photon photoelectron spectroscopy (XTPPS). The creation of deep single and double core vacancies induces significant reorganization of valence electrons. The corresponding relaxation energies and the interatomic relaxation energies are evaluated by CASSCF calculations. We propose a method how to experimentally extract these quantities by the measurement of single and double core-hole ionization potentials (IPs and DIPs). The influence of the chemical environment on these DIPs is also discussed for states with two holes at the same atomic site and states with two holes at two different atomic sites. Electron density difference between the ground and double core-hole states clearly shows the relaxations accompanying the double core-hole ionization. The effect is also compared with the sensitivity of single core hole ionization potentials (IPs) arising in single core hole electron spectroscopy. We have demonstrated the method for a representative set of small molecules LiF, BeO, BF, CO, N, CH, CH, CH, CO and NO. The scalar relativistic effect on IPs and on DIPs are briefly addressed.

I Introduction

The effect of the chemical environment manifests itself in energy differences of molecular core levels with respect to the atomic ones referred to as ”chemical shifts”. These can be measured by core level spectroscopies, e.g., by x-ray photoelectron spectroscopy (XPS) also known as electron spectroscopy for chemical analysis (ESCA) and by x-ray-induced Auger electron spectroscopy (XAES)Siegbahn71 (). Both spectroscopies have shown to be exceedingly successful tools to reveal the quantitative elemental composition of molecules and solids.

More than two decades ago, Cederbaum et al.Cederbaum86 (); Cederbaum87_1 () discovered that the creation of double core vacancies in molecular systems probes the chemical environment more sensitively than the creation of single core vacancies. Two-atomic site double ionization potentials, or briefly two-site DIPs ( or two-site double ionization energies, DIEs ) are particularly sensitive to the chemical environment as the examples of the CH, CH, CHCederbaum86 () and CHCederbaum87_1 () molecules demonstrate. The chemical shifts of one-atomic site DIPs, or briefly one-site DIPs, were found to be similar to the chemical shifts of the single core level ionization potentials (IPs), or ionization energies (IEs). This finding has given impetus to a number of theoretical studies aimed at elucidating properties of molecular double core hole statesCederbaum87_2 (); Ohrendorf91 (); Agren93 (); Reynaud96 ().

So far experimental explorations of double core hole states with conventional XPS were restricted to those having two vacancies on the same atomic site onlyKanter06 (); Kheifets09 () since the probability to produce a two-site double core hole state with one-photon absorption is practically zero at third-generation synchrotrons due to low x-ray intensities. This prevented further progress of the subject. The situation has changed with development of x-ray free-electron lasers (x-ray FELs)Feldhaus05 (). At FEL facilities in operations, such as FLASH in HamburgXFEL () and SPring-8 Compact SASE Source (SCSS) test acceleratorSCSS (), multi-photon absorption processes resulting in multiply ionized states of various systems have been extensively studiedSorokin06 (); Foehlisch07 (); Sato08 (); Jiang09 (); Fukuzawa09 (). In the x-ray FEL facility LCLS at SLAC National Accelerator Laboratory, which has just started operationsLCLS (), ultrashort pulses of a duration about 1-fs containing 2.4 photons with energies of 1 keV are expected to be generatedEmma04 (); Galayda () thus opening up the possibility to study molecular two-site double core hole states. Inspired by the advent of the x-ray FEL at LCLS, Santra et al.Santra09 () have demonstrated theoretically by the proof-of-principle simulations on the organic para-aminophenol molecule that two-site double core hole states can indeed be probed by means of x-ray two-photon photoelectron spectroscopy (XTPPS).

The operating principle of XTPPS is depicted schematically in Fig. 1. The initial step in XTPPS corresponds conventional XPS, i.e., a neutral molecule with an energy is irradiated by an x-ray photon with an energy and a photoelectron with the kinetic energy is ejected. This photoelectron carries information about a singly core ionized state of the molecule. If a second x-ray photon is absorbed before the intermediate core hole state decays, the second photoelectron expelled from the cation with the kinetic energy carries information about a double ionization potential. It is important to have an intense x-ray pulse with a duration that is significantly shorter than the core-hole lifetimes (typical lifetimes of core ionized states of F, O, N and C atoms are 3 to 7 femtoseconds). If the pulse duration is longer than these lifetime, then Auger decay is likely to occur prior to absorbing the second photon and thus the double core hole states may not be probed. A dicationic state of the system prepared by two-photon absorption decays electronically. Two primary Auger decays take place which overlap in time. An Auger decay happens preferably at that atomic site where the core hole has the shorter lifetime and an Auger electron with kinetic energy is ejected. This process proceeds in the presence of the second core hole which also decays via the Auger mechanism emitting an electron with kinetic energy . The electrons ejected via such a cascade of Auger decays can in principle be measured by a novel Auger spectroscopy which we call x-ray two-photon-induced Auger electron spectroscopy (XTPAES).

It is worthwhile to note that double core ionization can be accompanied by various shake-up processes similar to single core ionization. These many-body effects should manifest themselves in XTPPS spectra as satellites which are of interest as well. Both x-ray two-photon-induced Auger spectra and satellites structures will be addressed elsewhere.

The subject of the present paper is the main double core hole states. In order to provide a guideline for XTPPS experiments, we have performed ab initio calculations of core level single and double ionization potentials of LiF, BeO, BF, CO, N, CH, CH, CH, CO, and NO molecules. In addition we have explored the sensitivity of the DIPs to the chemical environment of the core ionized atoms. We decompose the DIPs in three physical contributions such as the orbital energies, the electrostatic repulsion energy between two core holes and the generalized relaxation energy and describe how the latter can be extracted from the experimental XTPPS spectra.

Ii Computational methods

Ab initio calculations of the vertical ionization potentials of the single and double core vacancy states of LiF, BeO, BF, CO, N, CH, CH, CH, CO, and NO were performed using the SCFBagus65 () and CASSCFWerner85 () methods. The molecular geometries used in these calculations were optimized at the Møller-Plesset level of theory (MP2) Hampel92 () employing the correlation-consistent polarized valence triple zeta (cc-pVTZ) basis sets of DunningDunning89 (). Depending on whether singly or doubly ionized states were considered, the configurations in the CASSCF method were restricted to those having one or two holes in the K-shell orbitals, respectively. We used the active spaces comprising all the occupied molecular orbitals (except for the 1 orbitals of the atoms other than H) and all the valence unoccupied and ones which contain large contributions from different atomic orbitals. Thus, the active space of the CASSCF calculations consists of , , , and orbitals with core occupancy being fixed. The cc-pVTZ basis sets were employed in all the CASSCF and SCF calculations. For CO and CH, the cc-pVDZ, cc-pCVTZ and cc-pVQZ basis sets were also used in order to examine the basis set dependence of our results. All the single and double core-hole states were solved by independent CASSCF calculations using different configuration space and, therefore, the calculated states are not strictly orthogonal to each other. However, the one and two-site double core-hole states are well separated in energy and their interaction are expected to be negligible.

For molecules with equivalent atoms, N, CH, CH, CH and CO, we calculated double ionization potentials using both localized and delocalized molecular orbital pictures following the recipe given by Cederbaum et al. Cederbaum86 () and discuss differences between them. Note that only core orbitals were localized which was performed with the Boys methodBoys66 (). In the localized representation we obtained the ionization potentials of the one-site double core hole states S and S as well as of the two-site double core hole states SS. Carrying out calculations with wave functions described by the linear combinations SS gives rise to double ionization potentials in the delocalized picture. Differences between single ionization potentials arising due to applying localized and delocalized representations are not studied in the present paper because they have been discussed in detail beforeDenis75 (); Cederbaum77 ().

In this work, we ignore the geometry relaxation of ionized state. In core ionization the change of geometry can be significant, depending on the case under investigation. In single core ionization one can explain the measurements well by employing the concept of vertical transitions. As in single ionization, also in XTPPS where the two X-ray photons must be absorbed within a time shorter than the Auger decay times, the concept of vertical transitions can be expected to be very useful.

In order to assess the impact of scalar relativistic effects on the core level single and double ionization potentials we made relativistic CASSCF calculations for the CO and BF molecules using the eighth order Douglas-Kroll-Hess Hamiltonian (DKH8)Douglas74 (); Hess85 (); Hess86 (); Reiher04_1 (); Reiher04_2 (). To get insight into the dynamic correlations, we also performed CI calculations with the CAS space plus single excitations from the CAS for both single and double core-hole states.

All calculations were done with the Molpro2008 quantum chemistry packageMOLPRO ().

Iii Results and discussions

iii.1 Single core hole states

Let us first discuss single core hole IPs. The ionization potential for the formation of a vacancy can be represented as

(1)

where is the corresponding orbital energy and is a contribution to the ionization potential due to relaxation and correlation effects:

(2)

The relaxation and electron correlations intermix with each other and cannot be strictly separated. The separation of these quantities was discussed in details in a perturbative way CeDoSc1980 () and in a nonperturbative way using MRCC DaMu2009 (). The correlation contribution can be further decomposed into two parts C1 and C2 (see Refs. Pickup73, and Cederbaum77, ) where C1 describes a part of the ground state pair correlation energy disappearing upon removal of an electron from the spin orbital , and C2 accounts for changes in the remaining pair correlation energy due to relaxation. Except for C1 which is a very small contribution, all contributions to are thus associated with relaxation of molecular orbitals. Therefore, for brevity of discussion, we may call the generalized relaxation energy.

A straitforward way to obtain the relaxation energy is to perform SCF calculations. is then derived as the difference between the respective orbital energy (with opposite sign) and the calculated IP. In order to get a correlation contribution to IP, post-Hartree-Fock calculations are generally needed. CASSCF is one of these methods. Noteworthy, in systems with core holes delocalized due to symmetry requirements, C2 can be accounted by performing SCF calculations using localized orbitals instead of delocalized ones as shown by Cederbaum and DomckeCederbaum77 ().

In Table 1 we list IPs calculated with the CASSCF and SCF methods together with available experimental valuesBakke80 (); Schirmer87 (); Kempgens97 (); Ehara06_1 (); Ehara07 (); Ehara06_2 (); Hatamoto07 (). Table 1 also contains the constituting parts of IPs, namely the orbital energies, the relaxation energies obtained from SCF calculations, as well as the generalized relaxation and pure correlation contributions, both obtained from CASSCF calculations. The correlation contributions were calculated by subtracting the CASSCF IPs from the SCF ones. Note that, since SCF calculations for molecules with equivalent atoms were performed using the localized representation, the calculated IPs correspond to the localized orbitals rather than to the delocalized 1 and 1.

In general, the agreement between the CASSCF and experimental results is reasonable. This concerns both the absolute values of IPs and the g-u energy splittings for molecules with equivalent atoms. Except for basis set effects which always are an issue in ab initio calculations, and relativistic effects, deviations from the experiment are attributed to the lack of dynamic correlations in the ground and single core hole states, and to the core-valence separation approximation employed in the calculations. We notice that influences of the above-mentioned effects and approximations partially compensate for each other. Indeed, performing calculations without core-valence separability lowers IPsAngonoa87 (). A lowering of IPs can also be achieved by improving basis sets. On the other hand, taking into account relativistic effects increases IPs. In Appendices A and B we explore the basis sets and relativistic effects in more detail.

It is interesting to compare the different contributions to IPs in Eq. (1). After the orbital energy, the relaxation energy represents the largest constituent part of a core level IP. It increases nearly proportional to the atomic number . For some molecules, however, remarkable deviations from this trend occur under influence of the chemical environment. A crucial role for the relaxation energy plays the change of the electron density distribution of valence electrons, , in an atom due to a formation of chemical bonds with neighbors, and the interaction of with the core hole. Ionic bonds give rise to the strongest changes of the electron density distribution. As a consequence, the relaxation energies associated with core ionization of electron acceptors in ionic molecules (e.g. O and F in BeO and LiF, respectively) are noticeably larger than the relaxation energies of the same atoms bound by covalent bonds with their neighbors (O and F in CO and BF, respectively). For other factors influencing the relaxation energies see Ref. Cederbaum86, .

In comparison to relaxation effects, correlation effects induced by core ionization are rather small. According to our calculations, the magnitude of the static correlation effects does not exceed 3 eV for the molecules studied and accounting for missing dynamic correlation can hardly modify this situation dramatically. Interestingly, the largest correlation effects manifest themselves in atoms whose neighbors are the strong electron acceptors O and F.

The effect of the chemical environment on core level ionization potentials of various systems including the molecules explored here is rather well established and we therefore refrain from long discussions on this subject. We only mention that the chemical environment is able to introduce large changes in the ionization potentials as, for example, can be realized by comparing molecules with ionic and covalent bonds. On the other hand, in the sequence of the CH, CH, CH molecules characterized by the triple, double and single carbon-carbon bond, respectively, the impact of the chemical environment is rather moderate. In contrast to single core hole ionization potentials, double core hole ionization potentials reveal much more pronounced sensitivity to the chemical environment as it was first demonstrated in Refs. Cederbaum86, and Cederbaum87_1, .

iii.2 Double core hole states

iii.2.1 General equations and results

In analogy to Eq. (1), we represent the double ionization potential of a state with two core vacancies and as

(3)

where is the repulsion-exchange energy of the two core holes. For an one-site double core hole state, it is described by the two-electron integral , or , and, for a two-site double core hole state, by a linear combination of the integrals and where the exchange term is negligibly small when the core holes are well localizedOhrendorf91 ().

The generalized relaxation can be decomposed into three parts

(4)

where given by Eq. (2) describes relaxation and correlation effects induced by creation of the core vacancy as there were no core vacancy . The relaxation and correlation energies are expected to be non-additive upon creation of multiple vacancies. A possible deviation from additivity is thus described in Eq. (4) by the non-additivity term .

Depending on whether two core holes were created on the same atomic site or on different atomic sites, may be called the excess generalized relaxation energy, , or the interatomic generalized relaxation energy, . Note that, while the and measure local properties of a core ionized atom, measures the impact of the environment ”between” the atoms involved.

In Table 2 we list the calculated double ionization potentials. We also show the correlation contributions to DIPs. As one can see these contributions are remarkably larger than those to the single IPs and may constitute 5.6 eV. In the special cases, however, when we performed calculations with delocalized core orbitals, differences between SCF and CASSCF values rise to 27-35 eV resulting from the failure of the SCF method in the delocalized picture to account for all relaxation contributions as described by Cederbaum et al.Cederbaum86 (); Cederbaum87_1 ().

One can notice by comparing Table 1 and  2 that the impact of the chemical environment is different for double and single ionization potentials. Of particular interest is to compare two-site DIPs with single IPs since their sensitivities to the chemical environment reveal major differences. A prominent example already discussed in detail in Ref. Cederbaum86, is the hydrocarbons CH, CH and CH. Here, the chemical shifts in the two-site DIPs are much more pronounced than in the single IPs being also attributed to different carbon-carbon bondlengths resulted from a different number of hydrogen atoms in these molecules. In XPS one can hardly distinguish between these three compounds while in XTPPS this should be possible in principle. The situation is somewhat different for the individual molecule NO. In this molecule the sensitivity of two-site DIPs to the chemical environment is lower compared to that of single IPs. Indeed, the core ionization potentials of the terminal and central nitrogen atoms differ by 4 eV whereas the difference between the N1sO1s and N1sO1s double ionization potentials constitutes 2.3-2.8 eV. The latter energy difference is much lower than 11 eV which one would expect taking into account only the differences between the NO and NO bondlengths and between the single core hole ionization potentials. As we show below, the reason for such a dramatic reduction of the chemical shift has to do with distinct relaxation processes induced by the creation of different pairs of core holes.

Taking into account Eqs. (1) and (4), we can represent as

(5)

and define the ionization potential of the core vacancy in the presence of the core vacancy as

(6)

whereas

(7)

is defined as the ionization potential of the core vacancy in the presence of the core vacancy .

Both the ionization potentials of a neutral system and the ionization potentials of a core-ionized one can be obtained experimentally, e.g., by means of XTPPS. In XTPPS, the kinetic energy of the first photoelectron ejected from the orbital defines , whereas the kinetic energy of the second photoelectron ejected from the orbital defines . Obviously, the sum gives . As shown below, important properties of the system under study can be extracted also from the measurable energy difference

(8)

Similar to , the kinetic energy depends significantly on the mutual arrangement of the core vacancies and in a molecule. This is clearly seen from Table 3 where we collect the kinetic energies of all the core electrons of the CO molecule which one would detect in an XTPPS experiment given that the molecule is irradiated by an x-ray pulse with photon energies of 1 keV. First of all, we notice that it is more difficult to remove an electron from the core ionized CO molecule than from the neutral one. The respective energy difference is about 70-90 eV when the first and the second core electrons are ejected from the same core orbital. This energy difference reduces drastically to about 15 eV when different core orbitals are affected. Apparently, the electrostatic interaction between the two core holes plays a crucial role here. exerts an influence on the above energy differences too, as can be deduced from Eqs. (6) and (7).

iii.2.2 One-site double core hole states

If then takes the form

(9)

We calculated for the molecules under study using the respective CASSCF single and double core hole ionization potentials and collect them in Table 4. The dependence of on the atomic number is displayed in Fig. 2(a).

The excess generalized relaxation energy can be easily obtained by measuring the energy difference provided that the integral is known. can be extracted from ab initio Hartree-Fock calculations on the electronic ground state of neutral molecules. Alternatively, it can be calculated by using the approximate analytical expression suggested in Ref. Cederbaum86, :

(10)

The respective results for and a difference between them are discussed in Appendix C.

Using Eq. (4) we represent as

(11)

It has been shown in Ref. Cederbaum86, that at the second order perturbation theory the following relationship between the relaxation energies is valid:

(12)

Since the impact of correlation into ionization potentials is small compared to the impact of relaxation, we expect that a similar relationship exists between the generalized relaxation energies and . Let us therefore introduce that

(13)

and find the optimal . After the substitution of Eqs. (11) and (13) into (9), we get

(14)

Now we can easily calculate by using the ab initio results for , and . The respective values of as a function of the atomic number are shown in Fig. 3. As one can see, deviations of the calculated from the expected value of 4 are rather small (15% in the worst case of Li) and therefore can be considered as a plausible approximation for the molecules studied in the present work.

As a result, we obtain the following expression for the generalized relaxation energy:

(15)

The values of calculated by means of Eq. (15) are given in Table 4 and also plotted in Fig. 2(b) as a function of . It is worthwhile to note a reasonable agreement between them and the corresponding results from Table  1.

iii.2.3 Two-site double core hole states

If then takes the form

(16)

where the repulsion-exchange energy has been approximated by the inverse of the distance between the two atoms with the core vacancies and . We calculated using the CASSCF double and single core hole ionization potentials and collected these results in Table  4.

By looking at Eq. (16), one expects a linear dependence between and . This expectation is not realized however as seen from Fig. 4(a) where a variation of with is shown. To elucidate the complex behavior of we calculated the interatomic generalized relaxation energy from Eq. (16) and plotted these results as a function of in Fig. 4(b). We found both positive and negative values of (see also Table  4) which indicate on an enhancement or suppression of relaxation effects, respectively.

In the case of diatomic molecules, is always negative and thus the relaxation is suppressed. This suppression may be interpreted by considering the change of the electron density. A core hole attracts valence electrons and increases the electron density in its vicinity, yielding a deficiency of the electron density in the vicinity of the atom with a core orbital . The relaxation energy corresponding to the creation of the core vacancy in the presence of the core hole is therefore reduced. The amplitudes of are smaller for LiF, BF, and N than for BeO, and CO. This is because the electrons are strongly attracted to the F site in LiF and BF, or tightened in the triple bond in N and thus the electron density flow due to core hole creation is suppressed by these chemical environments.

Values of for the polyatomic molecules CH, CH, and CH are, in contrast, positive. The enhancement of the relaxation for these molecules occurs due to the electron density flowing from the C-H bonds to the two carbon core hole sites.

The interatomic generalized relaxation energy exhibits a very interesting behavior with in triatomic molecules as seen from Fig. 4(b). is positive for CO and NO when and are in adjacent atoms, namely in C and O in CO and in N and N or in N and O in NNO. In these cases, the third atom plays the role of an electron donor and enhances the relaxation of the double core hole in the other two sites. The values of are, on the other hand, negative for CO and NO with two holes in the terminal atoms, namely with holes in the two O sites of CO and with holes in the N and O sites of NO. In these cases, the creation of the core hole on one site already withdraws the electron density from the central atom and thus reduces the possibility of relaxation due to the creation of the second hole in the other terminal site.

In order to analyze the reorganization caused by double core hole ionization, we calculated the electron density difference between the ground and double core hole ionized states. These electron density differences without the 1s contribution are plotted in Fig. 5, to better visualize the reorganization of the valence electrons. In the blue or green region, the electron density increases, while the density decreases in the red region. In the C state of CH, the electron density of the C-C, C-H bond and H atoms connected to C atom reduces and flows to the region around C atom as shown in Fig. 5(a). In the CC state of CH, on the other hand, the electron density in the region of C-H bonds and H atoms flows to the region of both C atoms as in Fig. 5(b). This explains the positive value of in the CC state as noted above. In the case of the O state of CO, the electron density of the CO bond is used for the reorganization around the O atom (Fig. 5(c)).

Iv Conclusions

We have computed the ionization potentials of single and double core hole states of the small molecules LiF, BeO, BF, CO, N, CH, CH, CH, CO, and NO by means of the SCF and CASSCF methods in order to explore the impact of the chemical environment on the respective ionization processes and provide a guidance for x-ray two-photon photoelectron spectroscopy (XTPPS) experiments.

Our calculations have demonstrated that except for NO, the double ionization potentials, especially the two-site ones of all these molecules are more sensitive to the chemical environment than the single ionization potentials. The sensitivity to the chemical environment of the two-site DIPs of NO is mainly governed by the interatomic relaxation energies which, in turn, strongly depend on the electron density distribution between the core-ionized atoms.

The quantities extracted from XTPPS are differences between the kinetic energies of core electrons ejected via the first and second ionization steps, i.e., of core electrons ejected from neutral and core-ionized systems, respectively. These kinetic energy differences are defined by a localization of the two core vacancies created and by relaxation processes induced by double core ionization. We have shown how one can extract the generalized relaxation energy associated with single core ionization as well as the excess and interatomic generalized relaxation energies associated with one-site and two-site double core ionizations, respectively, from experimental data by knowing the repulsion energy between the two core holes. The corresponding XTPPS experiments are now in preparation in the x-ray free electron laser facility LCLS at SLAC National Accelerator Laboratory.

Acknowledgements.
M.E. acknowledges the support from JST-CREST and a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science, the Next Generation Supercomputing Project, and the Molecular-Based New Computational Science Program, NINS. H.F. and K.U. acknowledge the support for the X-ray Free Electron Laser Utilization Research Project of Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). C.B. was supported by National Science Foundation under the grants No PHY-0701372 and No PHY-0449235. The Heidelberg group is grateful for financial support by the Deutsche Forschungsgemeinschaft. The computations were partly performed using the Research Center for Computational Science, Okazaki, Japan.

Appendix A Scalar relativistic effects

It is essential in view of their large impact to take into account relativistic effects when considering systems with heavy atoms. For light atoms, relativistic effects are of less importance but still should be accounted when highly accurate results are needed. In the present work we assess the impact of scalar relativistic effects on IPs and DIPs by carrying out calculations with the relativistic DKH8 Hamiltonian for the CO and BF molecules. The results of these calculations are shown in Table  5. As one can see the magnitude of the scalar relativistic effects on the single ionization potentials grows with the increasing atomic mass constituting 0.03, 0.09, 0.35, and 0.59 eV in the case of the B, C, O and F atoms, respectively. One of us has shown that a similar tendency is observed also for the third-row Si, P, S, Cl atomsEhara09 (). A growth of scalar relativistic effects with the atomic mass exhibits also in the case of double core hole ionization. We note that the relativistic effects on the one-site DIPs are about 2.3-3 times larger than those on the respective single IPs. Interestingly that in the case of two-site doubly ionized states the relativistic effects are perfectly described by the sum of the relativistic effects associated with the constituting single core vacancies.

Appendix B Basis set effects

In this section we explore the basis set dependence of the single and double core hole ionization potentials by examples of the CH and CO molecules. We have examined four basis sets: the Dunning’s correlation-consistent basis sets cc-pVXZ (X=D,T,Q) and the cc-pCVTZ one. The latter contains tight basis functions which are added for a better description of properties of core-level states. The results are collected in Table 6. By looking at the sequence of the cc-pVXZ results we find significant differences between the cc-pVDZ and cc-pVQZ values, especially in the case of the one-site DIPs where differences of nearly 10 eV are present. The major changes occur upon improving the basis set from a double- to triple-zeta quality. Choosing the cc-pVQZ basis sets lowers ionization potentials by only 0.2-1.2 eV relative to the cc-pVTZ values. The cc-pCVTZ results are lower in energy than the cc-pVTZ ones whereby they nearly coincide with the cc-pVQZ results in the case of the CO molecule. We used the cc-pVTZ basis throughout as a compromise between the accuracy of the results and the computational costs. For the double core-hole states, the relaxation of valence orbitals is important, in particular for the one-site states where all 1s electrons on one atom are missing.

Appendix C Evaluation of the integral

The approximate analytical expression (10) was proposed in Ref. Cederbaum86, for the two electron integral . By comparing the analytical results with the explicit ab initio ones (dotted curve and filled circles in Fig. 6, respectively) we found a progressively growing deviation between them with increasing (1.5 eV for lithium, 4.5 eV for fluorine). This deviation can be removed by substituting 1.037Z instead of Z in Eq. (10), where the coefficient 1.037 has been extracted from a linear fit (dashed curve in Fig. 6) of the ab initio results.

Appendix D Effect of dynamic correlations

We performed the CI calculations with the space of the singly excited configurations from the CASSCF configurations and examined the semi-internal correlation. The results for the single and double core-hole states of CH and CO were summarized in Table 7. The difference between the results of CI and CASSCF provides the effect of the semi-internal correlation. The semi-internal correlation has small effect on IPs of the single hole states, less than +0.03 eV for C1s and +0.25 eV for O1s. The effect for the one-site double core-hole states is significant as +0.24 +0.54 eV, while it is small for the two-site double core-hole states as +0.08 +0.12 eV.

References

  • (1) K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P. F. Hedén, K. Hamrin, U. Gelius, T. Bergmark, L. O. Werme, R. Manne and Y. Baer, ESCA applied to free molecules, North-Holland Publishing Company, Amsterdam, 1971.
  • (2) L. S. Cederbaum, F. Tarantelli, A. Sgamellotti and J. Schirmer, J. Chem. Phys. 85, 6513 (1986).
  • (3) L. S. Cederbaum, F. Tarantelli, A. Sgamellotti and J. Schirmer, J. Chem. Phys. 86, 2168 (1987).
  • (4) L. S. Cederbaum, Phys. Rev. A 35, 622 (1987).
  • (5) E. M.-L. Ohrendorf, L. S. Cederbaum and F. Tarantelli, Phys. Rev. A 44, 205 (1991).
  • (6) H. Ågren, H. Jørgen and Aa. Jensen, Chem. Phys. 172, 45 (1993).
  • (7) C. Reynaud, M.-A. Gaveau, K. Bisson, Ph. Millié, I. Nenner, S. Bodeur, P. Archirel and B. Lévy, J. Phys. B: At. Mol. Opt. Phys. 29, 5403 (1996).
  • (8) E. P. Kanter, R. W. Dunford, B. Krässig, S. H. Southworth and L. Young, Radiat. Phys. Chem. 75, 2174 (2006).
  • (9) A. K. Kheifets, J.-C. Dousse, M. Berset, I. Bray, W. Cao, K. Fennane, Y. Kayser, M. Kavcic, J. Szlachetko and M. Szlachetko, Phys. Rev. Lett. 102, 073006 (2009).
  • (10) J. Feldhaus, J. Arthur and J. B. Hastings, J. Phys. B: At. Mol. Opt. Phys. 38, S799 (2005).
  • (11) M. Altarelli, R. Brinkmann, M. Chergui, W. Decking, B. Dobson, S. Düsterer, G. Grübel, W. Graeff, H. Graafsma, J. Hajdu, J. Marangos, J. Pflüger, H. Redlin, D. Riley, I. Robinson, J. Rossbach, A. Schwarz, K. Tiedtke, T. Tschentscher, I. Vartaniants, H. Wabnitz, H. Weise, R. Wichmann, K. Witte, A. Wolf, M. Wulff, M. Yurkov, The Technical Design Report of the European XFEL, DESY XFEL Project Group, Deutsches Elektronen-Synchrotron (DESY), report No DESY 2006-097, Notkestraße 85, 22607 Hamburg, Germany, July 2006.
  • (12) T. Shintake, H. Tanaka, T. Hara, T. Tanaka, K. Togawa, M. Yabashi, Y. Otake, Y. Asano, T. Bizen, T. Fukui, S. Goto, A. Higashiya, T. Hirono, N. Hosoda, T. Inagaki, S. Inoue, M. Ishii, Y. Kim, H. Kimura, M. Kitamura, T. Kobayashi, H. Maesaka, T. Masuda, S. Matsui, T. Matsushita, X. Marechal, M. Nagasono, H. Ohashi, T. Ohata, T. Ohshima, K. Onoe, K. Shirasawa, T. Takagi, S. Takahashi, M. Takeuchi, K. Tamasaku, R. Tanaka, Y. Tanaka, T. Tanikawa, T. Togashi, S. Wu, A. Yamashita, K. Yanagida, C. Zhang, H. Kitamura and T. Ishikawa, Nature Photon 2, 555 (2008); T. Tanaka, T. Shintake, SCSS X-FEL conceptual design report, RIKEN Harima Institute/SPring-8, 1-1-1, Kouto, Mikazuki, Sayo, Hyogo, Japan 679-5148, May 2005.
  • (13) A. A. Sorokin, S. V. Bobashev, K. Tiedtke and M. Richter, J. Phys. B: At. Mol. Opt. Phys. 39, L299 (2006).
  • (14) A. Föhlisch, M. Nagasono, M. Deppe, E. Suljoti, F. Hennies, A. Pietzsch and W. Wurth, Phys. Rev. A 76, 013411 (2007).
  • (15) T. Sato, T. Okino, K. Yamanouchi, A. Yagishita, F. Kannari, K. Yamakawa, K. Midorikawa, H. Nakano, M. Yabashi, M. Nagasono and T. Ishikawa, Appl. Phys. Lett. 92, 154103 (2008).
  • (16) Y. H. Jiang, A. Rudenko, M. Kurka, K. U. Kühnel, T. Ergler, L. Foucar, M. Schöffler, S. Schössler, T. Havermeier, M. Smolarski, K. Cole, R. Dörner, S. Düsterer, R. Treusch, M. Gensch, C. D. Schröter, R. Moshammer and J. Ullrich, Phys. Rev. Lett. 102, 123002 (2009).
  • (17) H. Fukuzawa, K. Motomura, X.-J. Liu, G. Pruemper, M. Okunishi1, K. Ueda, N. Saito, H. Iwayama, K. Nagaya, M. Yao, M. Nagasono, A. Higashiya, M. Yabashi, T. Ishikawa, H. Ohashi and H. Kimura, J. Phys. B: At. Mol. Opt. Phys. 42, 181001 (2009).
  • (18) J. Arthur, P. Anfinrud, P. Audebert, K. Bane, I. Ben-Zvi, V. Bharadwaj, R. Bionta, P. Bolton, M. Borland, P. H. Bucksbaum, R. C. Cauble, J. Clendenin, M. Cornacchia, G. Decker, P. Den Hartog, S. Dierker, D. Dowell, D. Dungan, P. Emma, I. Evans, G. Faigel, R. Falcone, W. M. Fawley, M. Ferrario, A. S. Fisher, R. R. Freeman, J. Frisch J. Galayda, J.-C. Gauthier, S. Gierman, E. Gluskin, W. Graves, J. Hajdu, J. Hastings, K. Hodgson, Z. Huang, R. Humphry, P. Ilinski, D. Imre, C. Jacobsen, C.-C. Kao, K. R. Kase, K.-J. Kim, R. Kirby, J. Kirz, L. Klaisner, P. Krejcik, K. Kulander, O. L. Landen, R. W. Lee, C. Lewis, C. Limborg, E. I. Lindau, A. Lumpkin, G. Materlik, S. Mao, J. Miao, S. Mochrie, E. Moog, S. Milton, G. Mulhollan, K. Nelson, W. R. Nelson, R. Neutze, A. Ng, D. Nguyen, H.-D. Nuhn, D. T. Palmer, J. M. Paterson, C. Pellegrini, S. Reiche, M. Renner, D. Riley, C. V. Robinson, S. H. Rokni, S. J. Rose, J. Rosenzweig, R. Ruland, G. Ruocco, D. Saenz, S. Sasaki, D. Sayre, J. Schmerge, D. Schneider, C. Schroeder, L. Serafini, F. Sette, S. Sinha, D. van der Spoel, B. Stephenson, G. Stupakov, M. Sutton, A. Szöke, R. Tatchyn, A. Toor, E. Trakhtenberg, I. Vasserman, N. Vinokurov, X. J. Wang, D. Waltz, J. S. Wark, E. Weckert, Wilson-Squire Group, H. Winick, M. Woodley, A. Wootton, M. Wulff, M. Xie, R. Yotam, L. Young, A. Zewail, Linac coherent light source (LCLS): Conceptual design report, report No SLAC-R-593 and UC-414, April 2002.
  • (19) P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov and D. Walz, Phys. Rev. Lett. 92, 074801 (2004).
  • (20) J. Galayda, (private communication).
  • (21) R. Santra, N. Kryzhevoi and L. S. Cederbaum, Phys. Rev. Lett. 103, 013002 (2009).
  • (22) P. S. Bagus Phys. Rev. 139 A619 (1965).
  • (23) H.-J. Werner and P. J. Knowles, J. Chem. Phys. 82, 5053 (1985).
  • (24) C. Hampel, K. Peterson and H.-J. Werner, Chem. Phys. Lett. 190, 1 (1992).
  • (25) T. H. Dunning Jr., J. Chem. Phys. 90, 1007 (1989).
  • (26) S. F. Boys, in Quantum Theory of Atoms, Molecules and the Solid State, edited by P. O. Löwdin, (Academic Press, New York, 1966) p. 253.
  • (27) A. Denis, J. Langlet and J. P. Malrieu, Theor. Chim. Acta 38, 49 (1975).
  • (28) L. S. Cederbaum and W. Domcke, J. Chem. Phys. 66, 5084 (1977).
  • (29) M. Douglas and N. M. Kroll, Ann. Phys. 82, 89 (1974).
  • (30) B. A. Hess, Phys. Rev. A 32, 756 (1985).
  • (31) B. A. Hess, J. Chem. Phys. 33, 3742 (1986).
  • (32) M. Reiher and A. Wolf, J. Chem. Phys. 121, 2037 (2004).
  • (33) M. Reiher and A. Wolf, J. Chem. Phys. 121, 10945 (2004).
  • (34) MOLPRO, version 2008, a package of ab initio programs, H.-J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schütz, P. Celani, T. Korona, A. Mitrushenkov, G. Rauhut, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, G. Hetzer, T. Hrenar, G. Knizia, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, K. Pflüger, R. Pitzer, M. Reiher, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, M. Wang and A. Wolf.
  • (35) L.S. Cederbaum, W. Domcke and J. Schirmer, Phys. Rev. A 22, 206 (1980).
  • (36) D. Datta and D. Mukherjee, J. Chem. Phys. 131, 044124 (2009).
  • (37) B. T. Pickup and O. Goscinski, Mol. Phys. 26, 1013 (1973).
  • (38) A. A. Bakke, A. W. Chen and W. L. Jolly, J. Electron Spectrosc. Relat. Phenom. 20, 333 (1980).
  • (39) J. Schirmer, G. Angonoa, S. Svensson, D. Nordfors, and U. Gelius, J. Phys. B: At. Mol. Opt. Phys. 20, 6031 (1987).
  • (40) B. Kempgens, H. Köppel, A. Kivimäki, M. Neeb, L. S. Cederbaum and A. M. Bradshaw, Phys. Rev. Lett. 79, 3617 (1997).
  • (41) M. Ehara, K. Kuramoto, H. Nakatsuji, M. Hoshino, T. Tanaka, M. Kitajima, H. Tanaka, Y. Tamenori, A. D. Fanis and K. Ueda, J. Chem. Phys. 125, 114304 (2006).
  • (42) M. Ehara, R. Tamaki, H. Nakatsuji, R. R. Lucchese, J. Söderström, T. Tanaka, M. Hoshino, M. Kitajima, H. Tanaka, A. D. Fanis and K. Ueda, Chem. Phys. Lett. 438, 14 (2007).
  • (43) M. Ehara, H. Nakatsuji, M. Matsumoto, T. Hatamoto, X.-J. Liu, T. Lischke, G. Pruemper, T. Tanaka, C. Makochekanwa, M. Hoshino, H. Tanaka, J. R. Harries, Y. Tamenori, K. Ueda, J. Chem. Phys. 124, 124311 (2006).
  • (44) T. Hatamoto, M. Matsumoto, X.-J. Liu, K. Ueda, M. Hoshino, K. Nakagawa, T. Tanaka, H. Tanaka, M. Ehara, R. Tamaki and H. Nakatsuji, J. Electron Spectrosc. Relat. Phenom. 155, 54 (2007).
  • (45) G. Angonoa, O. Walter and J. Schirmer, J. Chem. Phys. 87, 6789 (1987).
  • (46) M. Ehara, K. Kuramoto and H. Nakatsuji, Chem. Phys. 356, 195 (2009).
Molecule Core level,  - SCF CASSCF Expt.
IP IP
LiF Li1s 66.407 65.460 0.947 65.334 1.073 0.126
F1s  710.484  688.187  22.297  688.044  22.440  0.143
BeO Be1s 128.383 124.406 3.977 123.399 4.984 1.007
O1s 556.694 535.181 21.513 535.075 21.619 0.106
BF B1s 209.735 202.539 7.196 201.724 8.011 0.815
F1s 717.810 695.873 21.937 695.915 21.895 -0.042
CO C1s 309.111 298.256 10.855 296.358 12.753 1.898 296.069
O1s 562.348 542.801 19.547 542.820 19.528 -0.019 542.543
NO N1s 427.159 409.615 17.544 408.614 18.545 1.001 408.5
N1s 432.005 415.373 16.632 412.524 19.481 2.849 412.5
O1s 563.760 543.046 20.714 542.537 21.223 0.509 542.0
CO C1s 311.930 300.607 11.323 297.647 14.283 2.960 296.78
O1g 561.956 541.979 19.977 542.870 19.086 -0.891 540.6
O1u 561.954 19.975 542.868 19.086 -0.889
CH C1g 305.897 292.535 13.362 292.202 14.062 0.700 291.14, 291.20
C1u 305.794 13.259 292.111 14.054 0.795
CH C1g 305.557 291.801 13.756 291.344 14.213 0.457 290.70, 290.88
C1u 305.508 13.707 291.297 14.211 0.504
CH C1g 305.040 291.774 13.266 291.147 13.893 0.627 290.76, 290.74
C1u 305.023 13.249 291.125 13.898 0.649
N N1g 426.686 411.242 15.444 411.027 15.659 0.215 409.93
N1u 426.588 15.346 410.932 15.656 0.310 409.82
Table 1: Single core hole ionization potentials and their constituent parts (in eV) as calculated with the SCF and CASSCF methods. Experimental values of the ionization potentials, where available, taken from Refs. 38-44 are also shown. The cc-pVTZ basis sets were employed.
Molecule Core level SCF CASSCF
DIP DIP C
LiF Li1s 173.125 172.595 0.530
F1s 1480.418 1481.495 -1.077
Li1sF1s, S 763.447 763.211 0.236
Li1sF1s, T 763.443 763.277 0.166
BeO Be1s 300.585 298.032 2.553
O1s 1158.135 1159.351 -1.216
Be1sO1s, S 672.823 671.801 1.022
Be1sO1s, T 672.823 672.128 0.695
BF B1s 468.243 465.323 2.920
F1s 1494.930 1495.809 -0.879
B1sF1s, S 910.946 910.568 0.378
B1sF1s, T 910.946 910.678 0.268
CO C1s 667.902 664.418 3.484
O1s 1175.376 1176.561 -1.185
C1sO1s, S 857.072 854.743 2.329
C1sO1s, T 857.072 855.200 1.872
NO N1s 894.485 893.926 0.559
N1s 906.773 902.312 4.461
O1s 1173.683 1173.249 0.434
N1sN1s, S 838.282 832.962 5.320
N1sN1s, T 838.279 833.215 5.064
N1sO1s, S 965.806 963.041 2.765
N1sO1s, T 965.806 963.266 2.540
N1sO1s, S 968.082 965.793 2.289
N1sO1s, T 968.082 965.623 2.459
Table 2: Calculated double core hole ionization potentials and the static correlation energies C (in eV). T and S refer to triplet and singlet couplings of two core holes created on different atomic sites, respectively. The cc-pVTZ basis sets were employed.
CO C1s 670.280 664.629 5.651
O1s 1172.821 1171.909 0.912
C1sO1s, S 854.682 851.059 3.623
C1sO1s, T 854.682 851.199 3.483
O1sO1s, S 1094.795 1094.090 0.705
O1sO1s, T 1094.795 1094.167 0.628
CH C1s 651.265 650.228 1.037
C1s 651.265 650.228 1.037
C1sC1s, S 598.281 594.590 3.691
C1sC1s, T 598.281 595.197 3.084
C1s-C1s 681.646 651.334 30.312
C1s+C1s 681.646 651.334 30.312
CH C1s 648.964 648.556 0.408
C1s 648.964 648.556 0.408
C1sC1s, S 594.850 591.514 3.336
C1sC1s, T 594.850 591.956 2.894
C1s-C1s 679.386 649.703 29.683
C1s+C1s 679.386 649.703 29.683
CH C1s 649.714 648.827 0.887
C1s 649.714 648.827 0.887
C1sC1s, S 591.447 589.007 2.440
C1sC1s, T 591.447 589.192 2.255
C1s-C1s 677.339 649.898 27.441
C1s+C1s 677.339 649.898 27.441
N N1s 901.704 901.155 0.549
N1s 901.704 901.155 0.549
N1sN1s, S 839.912 835.784 4.128
N1sN1s, T 839.912 836.437 3.475
N1s-N1s 938.943 903.727 35.216
N1s+N1s 938.943 903.727 35.216
       State KE (eV)
C1s   703.642
C1sC1s 631.940
O1sC1s, S 688.077
O1sC1s T 687.620
O1s 457.180
O1sO1s 366.259
C1sO1s, S 441.615
C1sO1s, T 441.518
Table 3: Kinetic energies of photoelectrons one would detect in an XTPPS experiment by irradiating the CO molecule with an x-ray pulse with photon energies of 1 keV. The sequence of the core vacancies in the records reflects the sequence of ionization steps. T and S refer to triplet and singlet couplings of two core holes created on different atomic sites, respectively.
Molecule Energy difference Generalized relaxation energy
LiF (Li1s) 41.927 (Li1s) 1.41
(F1s) 105.407 (F1s) 20.17
(Li1s, F1s) 9.833 (Li1s, F1s) -0.74
BeO (Be1s) 51.234 (Be1s) 5.23
(O1s) 89.201 (O1s) 19.90
(Be1s, O1s) 13.327 (Be1s, O1s) -2.69
BF (B1s) 61.875 (B1s) 8.38
(F1s) 103.979 (F1s) 20.88
(B1s, F1s) 12.929 (B1s, F1s) -1.52
CO (C1s) 71.702 (C1s) 11.87
(O1s) 90.921 (O1s) 19.03
(C1s, O1s) 15.565 (C1s, O1s) -2.80
N (N1s) 79.196 (N1s) 17.65
(N1s, N1s) 13.825 (N1s, N1s) -0.65
CH (C1s) 66.653 (C1s) 15.06
(C1s, C1s) 11.015 (C1s, C1s) 0.96
CH (C1s) 65.915 (C1s) 15.42
(C1s, C1s) 8.873 (C1s, C1s) 1.94
CH (C1s) 66.555 (C1s) 15.06
(C1s, C1s) 6.735 (C1s, C1s) 2.72
CO (C1s) 69.335 (C1s) 13.76
(O1s) 86.171 (O1s) 22.70
(C1s, O1s) 10.572 (C1s, O1s) 1.79
(O1s, O1s) 8.352 (O1s, O1s) -2.19
NO (N1s) 76.698 (N1s) 17.78
(N1s) 77.264 (N1s) 17.47
(O1s) 88.175 (O1s) 20.40
(N1s, N1s) 11.827 (N1s, N1s) 1.11
(N1s, O1s) 10.732 (N1s, O1s) 0.07
(N1s, O1s) 11.890 (N1s, O1s) -6.00
Table 4: Calculated energy differences and together with the intra- and interatomic generalized relaxation energies and (in eV).
Molecule State Ionization Potential Difference
CO C 296.446 0.088
O 543.166 0.346
C 664.661 0.243
O 1177.408 0.847
CO, S 855.176 0.433
CO, T 855.632 0.432
BF B 201.758 0.034
F 696.500 0.585
B 465.425 0.102
F 1497.177 1.368
BF, S 911.187 0.619
BF, T 911.297 0.619
Table 5: Single and double core hole ionization potentials of CO and BF calculated with the CASSCF method using the relativistic DKH8 Hamiltonian and the differences between them and the respective non-relativistic results given in Tables 1 and 2. All energies are in eV. T and S refer to triplet and singlet couplings of two core holes created on different atomic sites, respectively.
Molecule State cc-pVDZ cc-pVTZ cc-pCVTZ cc-pVQZ
CH C 293.262 292.202 292.111 291.728
C 293.172 292.111 292.020 291.634
C 658.167 650.228 649.412 649.711
CC, S 598.350 594.590 594.337 593.908
CC, T 598.853 595.197 594.971 594.592
CO C 298.062 296.358 296.239 296.229
O 544.881 542.820 542.611 542.559
C 672.908 664.418 663.663 663.632
O 1184.846 1176.561 1175.469 1175.373
CO, S 859.146 854.743 854.452 854.425
CO, T 859.491 855.200 854.931 854.905
Table 6: Basis set dependence of the CASSCF single and double core hole ionization potentials. All energies are in eV. T and S refer to triplet and singlet couplings of two core holes created on different atomic sites, respectively.
Molecule State CASSCF CI
CH C 292.202 292.216
C 292.111 292.127
C 650.228 650.586
CC, S 594.590 594.728
CC, T 595.197 595.319
CO C 296.358 296.386
O 542.820 543.070
C 664.418 664.658
O 1176.561 1177.096
CO, S 854.743 854.833
CO, T 855.200 855.276
Table 7: Effect of dynamic correlations examined by the CI calculation. All energies are in eV.

FIGURE CAPTIONS

Figure 1: (Color online). Schematic picture of x-ray two-photon photoelectron spectroscopy (XTPPS) and x-ray two-photon-induced Auger electron spectroscopy (XTPAES). See text for explanations. In this picture it is assumed that the second photon is absorbed before Auger decay takes place and that one core hole decays much faster than the other. In reality all processes overlap.
Figure 2: (a) The energy difference as a function of the atomic number Z of the atom with the core orbital ; (b) The generalized relaxation energy calculated by means of Eq. (15) as a function of Z.
Figure 3: The ratio as a function of the atomic number Z of the atom with the core orbital .
Figure 4: (a) The energy difference as a function of the inverse distance between the atoms with the core orbitals and ; (b) The interatomic generalized relaxation energy calculated by means of Eq. (16) as a function of .
Figure 5: Differences between the valence electron density distributions of the ground and double core hole states: (a) C1s of CH; (b) C1sC1s (singlet) of CH; (c) O1s of CO
Figure 6: The two-electron integral as a function of the atomic number Z of the atom with the core orbital . The results extracted from the ab initio calculations (filled circles) are compared with those calculated by means of Eq. (10) (dotted curve). A linear fit of the ab initio results is also shown (dashed curve).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
187586
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description